How can I fix damp grout between floor tiles?

If you have damp grout on the floor tiles, you'll need to remove that tile and some others around it. Sounds to me that water from somewhere is getting to that low point in your floor. But before removing, have a quick look at the bottom of you basin, w/c and bath pipes, as there maybe a small leak happening.

Use ardex 2 and 1 grout also beforehand look for tiles moving and replace with rapid set and male sure all joint between tiles are dry

I would remove the tile and those immediately surrounding it to check if water is getting in somewhere.

The answer is you’d need to remove the tiles to find out what the issue is. Incorrectly laid and possibly no moisture matting used!