I often add solder to make it easier for the wick to get everything. If the original assy was Lead-Free, using low temp solder (I can has lead? As a treat?) may make a difference here as well. Flux pen on the solder wick also seems to help especially if your wick is kinda crusty.

A desoldering pump (manual model, $10 or so for a decent one) is very suitable for removing solder from through-holes, if that is the main issue.