I have two ideas,
one, you may need to replace the freewheel. I have a similar problem on a letour, the smallest rear cog skips under pressure.
two, if the chain has been replaced, it may be too long and the derailleur may not be able to take up all of the slack.
Anyways, those are my suggestions, but i'd wait for some more people to chime in as well
good luck!