Methods for removing solder from inside a header socket
If I understand you correctly, the solder is inside the contact, not the solder hole, then there is no practical way to fix it.

Some headers use a forked contact, others use a hollow square. In both cases, the solder gets wicked up with Capillary Action.

Once inside, the heat and time needed to reflow the solder, and wick it up with any typical desoldering wick or desoldering sucker tool will melt the plastic of the header. You won't be able to get close enough with most soldering irons to do it.

 

Desolder the entire header, and replace it with a new one, using less solder of course. It's the only proper and practical way to do it.