After removing the reader altogether, I would recommend adding a little bit of flux, and using solder wick, removing all old solder, and then taking a small blob of solder (mind the heat of the iron) rolling that ball of solder on the pads with sufficient amount of flux to tin the pads with new solder. Don't overdo this, or pads will come off.

Excessive solder has to be removed, though. Like that blob on the left hand side, next to the capacitor, which is remnants of desoldering. Sometimes it helps to replace the lead-free solder or just add more leaded solder to melt the joint.