specificity is usually such that the best thing you can do to improve x is to do x.
so you wanna powerclean more weight - do more powercleans
you wanna back squat more weight - do more back squats etc.
sometimes though there can be a weak link. some kind of assistance exercise might help you improve the main lift because it fixes up the weak link (e.g., i find that weighted planks strengthens my abdominals which results in a better front squat since i don't fold forwards before my legs give up). this kind of work relies on your having good movement patterns etc with the main lift, though.
best thing to help with the first pull is doing first pulls.
there is some controversy over the use of the back squat for olympic weightlifters. having a hooge back squat doesn't necessarily translate to cleaning or snatching a lot of weight (because of the technical nature of cleans and snatches). front squats are more similar to cleans (with the bar in the rack position) but similarly having a hooge front squat doesn't necessarily translate to cleaning a lot of weight (because of the technical nature of cleans).
people get pinned in the bottom of a clean with one hell of a lot less than their max front squat. because they didn't hit a tight bottom position when they dropped under and / or they aren't fast enough to ride the bounce out the hole. squat strength does not necessarily translate to big olympic lifts.
also... having strong powercleans and snatches doesn't necessarily translate to having a particularly strong full clean and snatch. because once again the technical demands are different. it is one thing to do a decent first and second pull... it is much much harder to add a fast decent into a tight bottom position cleanly riding the bounce out AFTER full extension... and the jerk is something else again...
so... in a technically proficient lifter... depending on their weaknesses... squatting might help their cleaning or snatching (if they have trouble with standing them up). powercleaning / powersnatching might help their cleaning or snatching (if they have trouble with extending fully / pulling the bar high enough to drop under). if one isn't technically proficient with the full lifts then focusing on squat strength or powerclean / powersnatch strength isn't likely to help the full lifts, though. what is most likely to help the full lifts is doing the full lifts. or doing partial versions of the full lifts (e.g., first pulls, power versions, squats) not with a 'move the most weight' mentality but rather with a 'developing technical proficiency in a part with an eye to coordinating it into technical proficiency with the whole' eye.
but opinions vary, i suppose.