What can I do about the large gap between my new 6-tooth bridge and my gum? Air whistles through, I can't say "s" properly.

Depends on how big the gap is.  Sometimes we could add bone/ soft tissue (ridge augmentation) to plump the site again.  you could see a periodontist for a consultation and explore surgical options.

It is very possible that you have had some bone loss in the area of the extraction which can lead to gaps in your bridge contact with the tissue.  And yes it can lead to speech problems.  Unfortunately the only way to fix this is to redo the bridge.  And you want to make sure, if this is done, that the bone has finished the major part of the remodeling process or you could be facing the same thing a few years down the road.  Good luck!