This problem is similar to a post of a few months ago, but not quite.

Here are the details.  88 Integra RS, 120k.  Recent air filter.  Plugs
not new but okay.  Could probably stand a cap and rotor.  It has great
difficulty getting started.  This problem has been getting steadily
worse over the last year or so.  It used to be only a problem on hot
days when the car was already warm.  Crank, crank, crank, and nothing.
Often, just shutting off the ignition and letting it sit a few
seconds, then retrying would be enough for it to start right up.  Now,
that does not even help.  Once in a while the car will actually start
right up with no problems  But this is rare now and pretty much only
happens after sitting overnight.  It's at the point now that all the
cranking required to get it started is nearly enough to run down the
battery.  IOW, it's not very driveable now.

I've tried all manner of oddball things to try to help it out, from
holding the gas pedal all the way to the floor while cranking to
pumping the gas, to keeping off the gas completely.  Nothing seems to
make any difference.  When the car finally does start, it's definitely
as though only one cylinder is firing for a few seconds.  It will
chug, chug, chug, and I often need to keep the starter running until
it gets chugging quickly enough to run on its own.  Then, all of a
sudden it's as though all the cylinders kick in at once, and it's up
and running, very nicely.

There is never any black smoke once it starts.  If anything, I'd say
it's not getting enough fuel upon startup.  If this were a carbureted
car, I'd say without a doubt that the accelerator pump is not
functioning propery, and it's just too lean.  This can also be evident
when driving and taking off at a green light, because sometimes the
car bogs and nearly dies for an instant, which is another sympton very
reminscent of a bad accelerator pump (if it were carbureted).
Otherwise, the car runs reasonably well and smooth once it's running.

I poked around the intake manifold and such, not sure what to look
for.  I did notice a little tang that appears to control the idle
speed looked bent out of whack and was not making any contact with the
throttle lever.  I tweaked it so that the car now idles around 1000rpm
in neutral.  It used to be about 600 which was just too low.  None of
this helped the starting problem in any way though.

What could the problem be?  Is there something similar to an
accelerator pump in EFI engines?  If so, that's the first place I'd
want to look.  If not, what else might be the trouble?  Is there some
way I can diagnose this better?

Thanks.