What about the oil in the dashpot?

The second capacitor may be for interference suppression.  Check the
connections. The normal capacitor (condenser) fits across the contact
points themselves (in other words, from the - side of the coil to
ground). An interference suppressing capacitor fits from the + side of
the coil to ground.

The hesitation might be because the dashpot oil is dry.  Unscrew the
damper cap and feel when the damper hits the oil as you put the rod back
in.  It should be 5 to 10mm before the threaded bit starts. (You'll
easily feel when the brass bit hits the oil, the resistance can be
felt).

Also check that the vacuum advance is working properly.

this sort of problem could be from a number of causes, ill list the ones ive
thought of, but im sure the group will discuss more...

1) as chris said it my be due to vac advance.  to test for leaky pipes,
unplg them all and suck one end while blocking the other with finger. if
leaky - replace.  if a pipe was leaking then more air would be entering the
engine after the carb, thus the engine is not getting correct fuel/aiir mix
even tho the carb may be set up correctly.  next (with engine off) suck on
the pipe going to the dizzy - if you can hear the plate clunking when u suck
and release then the dizzy advance is working

2) fueling ; is the car standard or does the engine have mods.  even a
different air filter could screw the performance up - i.e. say a k&n is
fitted without a proper fuel set up, the engine is getting a lot more air
than necessary and thus wont go anywhere when u put foot down.  if the
engine is getting too much fuel however, it doesnt know what to do with it -
i had this problem, at 2000rpm and floor it the engine cant cope and bogged
down till it got past 3000 and flew.  the only accurate way to set up the
fuel is on a rolling road, you could try altering the mixture yourself  (do
you know which screw it is?) and trialing it on the road

3) an air leak after the carb would have the same effect as in 1) check carb
and inlet gaskets for leaks.  with the engine running squirt wd40 all round
these gaskets.  the fluid temporarily seals the leaks in the gaskets and you
should hear the engine pick up.  do this on the take off point (think its
for the brake servo) - i had an air leak hear on a minispares inlet
manifold - i tightened it up as much as poss and it still leaked, so i ended
up putting a crushable sump plug washer in there to seal it up.

hope this helps
Ryan

Just a bit of an update guys,
Changed the points & condenser, and fitted a thermostat (there wasn't one in
there) the car seems better now at least it's reaching its working
temperature.
I'll leave it with my son to see if he thinks there is some improvement.

One other thing for now does anyone know if minis usually sound very throaty
when you first accelerate or may there be some leak (exhaust or manifold).
I'm going to see if I can spot anything tomorrow, thanks for your help so
far.

Neil
PS there are no mods to this car at all, though I'd mention that.