We had experience with sludge build-up on the oil pump intake screen in our Grand Marquis. The oil pan does not just come off, a cross-member has to be removed, or the engine removed, or you drill a hole in the pan and then weld it shut... but I digress.

So, we tried reaching through the oil drain hole to knock off as much as possible, and then an oil change with synthetic oil. We figured that since it was motor oil it could not hurt, and being a higher-detergent synthetic, it might help to clean out a lot of the gunk.

There were 2 or 3 episodes of low oil pressure warning lights. Each time we stopped, waited a moment, and re-started to find it was fine.

Once the valve train got really clackity and clattery due to a temporary clot in the system. This also cleared up after a stop and restart.

6,000 miles since, 217,000+ on the odometer, and it's doing fine. We are back to using dino oil since it consumes about a quart per 1000 mile due to worn valve seals.

To summarize, if you think that varnish is blocking a lifter, you could try a change with synthetic so see if it will dissolve.

Others will chime in here...