Here may be your problems, in an order of likelihood;

1. The cable is "binding" at its clip point on the speedo. Some mildly frustrating fiddling can cure that problem..it is related to the length of the cable tip entering the speedo. I use the reversible drill to check functioning here and after it works smoothly with the drill I clip it on the gearbox.

2. Bad routing.  Here is a picture of the original cable that graced the cars..and ALL Plus 8s from the introduction of the 5-speed in 1977 until the advent of the 3.9 dictated the need for a transducer. It sells for $30..where the later ones with a transducer have three parts(lower cable/transducer/upper cable) for a total of $170 IF you can find one. The Morgan cable is 53 inches long and a bad routing can create binding.

Morgan had two speedo routings. One went out the side of the bellhousing and then back in to the car through the firewall. Even at 53 inches this stretches the cable so tight that the cable will inevitably (5 minutes - 20 years) bind or snap the plastic press fit lock. The other routing...which is less "clean" but safer, has the cable coming out a hole (through a rubber grommet) in the top of the gearbox cover and then under the padded leathercover until it ends and then loops back into the speedo. This routing leaves more slack and less chance of binding.

3.  Dirty or worn cable sleeve. The cable sleeve is dirty, defective or has oil in it. It can sometimes be rectified by running the cable through ita few times to clean it. With the cable sleeve unclipped/removed at either end (nor necessary to do both) it is a simple matter to remove the cable from its sleeve, clean it, lubricate or replace it. I maintain mine once every year as I find it gives me a steady needle on the speedo. Lubricate with speedo cable lubricant or powdered graphite.

4.. The ends of the cable have worn. This problem normally means that the female entry point has worn as well. A worn piece wears everything it touches.

5. The speedo gear has gone. These little plastic gears are very cheap and, of course, no one who has them will supply them to you. This is another area where I would like to get a supply source up on GoMoG. The gear costs pennies, is easy to replace and saves a $125 USD or 90 pound service charge and two week turn-around time.

6. The speedo needle is stuck. On some speedos, for some bizarre reason, the needle can go too far
back to it is beyond zero and then it can jam itself at 0. This is easily remedied.  I just foxed the same problem on the e-type with its Smiths.

7. The gear box connection is ascew. When you get everything together, be very careful with the connection at the gearbox...you must have it perfectly placed and tight or the cable will not like you either.
If the small bracket at the gearbox entry is loose, it will give you a very nervous needle..with a sad up and down movement straddling 5 to 15mph.

8. Later cars have a speedo cable in three parts, a gearbox to transducer section, the transducer and the transducer to speedo.  The parts can be purchased in sections but are becoming difficult to find. You can have one made. see http://www.speedycables.com/Control_Cables.htm The transdicer sends a signal necessary for the smooth running of the engine to the ECU. If this section is broken or the transducer is fault you will have a red warning light and poor performance. Often, the transducer ground is at fault. Simply verify that and off you go.
 

ADDENDUM BY DUNCAN CHARLTON

While a common problem is routing the cable with too-sharp a bend in it, the internals of ours were the
problem and this requires removing the internals, which is not hard -- after rotating the faceplate ring, take off the glass on the front.  You might want to replace the rubber o-ring before reassembly.  Undo the two
screws at the back to get the internals out.

See the attached photo to see what the back portion looks like on a 1971 Plus 8, which should be similar to all of that era.  The significant parts on the input shaft are the brass odometer drive gear and the bar magnet, both of which spin at the speed of the speedo cable.  The flat aluminum plate with the turned-up edge is attached to the speedo needle.

The brass odometer drive gear is pressed onto the input shaft, and between the gear and the bar magnet is a bearing plate that's held to the casting by a couple of soft rivets.  The cable spins the input shaft, which spins the  magnet, which induces the aluminum plate to rotate slightly; the faster the input shaft turns, the more the force trying to move the aluminum plate, which is fighting the spring that returns the needle to zero when there is no motion of the input shaft. When the aluminum plate moves, the needle on the front of the instrument moves (don't ask me how a magnet affects aluminum -- I can't explain it).

Pressure from the speedo cable pushes inward on the driveshaft, and the wear point is the far side (front) of the brass gear, where it rides on the bearing plate.  With enough wear, the bar magnet makes intermittent contact with the flat aluminum plate, jerking the needle on the speedometer.  Backing out on the cable fitting to reduce pressure on the mechanism might help be removing this forward pressure.

The long term fix may be to take the speedo apart.  The second image shows the magnet side of the input shaft once it's removed from the casting.  I drilled out the rivets, removed the input shaft with gear and magnet attached, and used a socket and a small hammer to tap the brass drive gear closer to the bar magnet.  This has the effect of moving the bar magnet towards the back of the instrument, pulling it away from the aluminum plate slightly.  To reassemble it, I annealed some copper household wire (probably 12 gauge) "Romex" -- the stuff that's inside your walls.  I stripped out a piece of this solid conductor, heated it to carrot red and let it air cool.  This makes it very soft and easy to deform.  I cut a couple of pieces to the right length to make rivets, put them in the holes, and tapped down on the ends with a nail punch to flatten the ends. The assembly rivets attach to the bearing plate.  The casting attaches to the front portion of the assembly (the part with the odometer) from the front (odometer) side with small screws.

The pictures I have posted show the speedo unit disassembled to a greater degree than necessary to make this particular repair.  If there's any way to remove the small screws mentioned above without removing the faceplate (and therefore the speedo needle), I recomment doing that -- but you'd need some sort of tiny angled screwdriver.

This fix worked for our speedometer.  While I was in there, I noticed that a tiny spring on the pawl that advances the odometer was missing and took the opportunity to some general cleaning.  One mistake I made -- I used some very light oil (sewing machine oil) to lubricate things. I believe the little socket on the bar magnet, where the pointed tip at the back of the speedo needle drive shaft fits, should not be lubricated with oil, since the needle is reluctant to return to zero when the calibration is correct at 60 mph.

If you lost the needle placement position, the "correct" place for you to position it is with the needle pointing to the tiny dot next to the stop post, on the other side from the zero.  You're then expected to
flip the needle past the post so that there's slight spring tension holding it against the post.

Get a friend who has a car with an accurate speedometer to pace you at 60 mph, note the needle position.  If you're reading something different, you'll have to recalibrate, which is very simple.  Take the
guts out of the speedo again and rotate the needle to the speed it was indicating at a true 60 mph.  Grab the aluminum plate with your fingers in the back and simply twist the needle (at its hub, not at the tip) to
indicate 60 mph.  Reassemble it and you're done.

Duncan