This is the locking gib. It is used to keep Y axis in position and make machine more vibes resistant when U machine planes or grooves along X axis.
There can be another one on the other side for play reduction.

Normally the tightening gib should fix the problem with play on both sides of dovetail, but maybe if this machine worked a lot with large play matching surface could get waved so it won't work properly with direct contact.
Other possibility is that large wear of surfaces caused contact on non flat surfaces (on the edge of dove tail or on flat surface beneeth it) chamfering edges of matching surfaces could help then.