I find that skipping problems are rarely related to the derailleur itself. It is usually the shift levers, chain or freewheel.
The fact that it occurs on the 4th gear makes me think that the problems is related to the cog itself. It could be a damaged tooth (bent, chipped or burred). However, it sounds like the bicycle has probably been ridden in the 4th cog for most of it's life and that cog is excessively worn. When a new chain is installed it meshes properly with the relatively unused gears but poorly with the highly used cog.
It could also be that the shift lever tension screw has backed-off but this generally affects all gears. However, this is the easiest "fix", so check the torque anyway. It may take a bit of fine tuning to have the screw tight enough to maintain lever position but not be too hard to move the lever.
It could be a stiff link in the chain, but this generally is worst on the smallest cog, as requires the links to bend the most.
I suggest you spin the crank slowly to see where the skip is happening. Mark the cog and chain with a dab of paint, marker or tape. See it repeats in the same spot, either on the cog or chain. If it does you have a specific problem with a tooth or link respectively. If not, it's a general wear problem and most likely the freewheel, especially if the chain has been replaced.