Pat - Fothergillas do not require pruning but for sure you can prune them to keep them at the size at the appropriate size for your area.  That being said, the best time to lightly prune them is immediately after flowering.  You can prune out dead branches at any time.  Because Fothergillas bloom on the previous year's growth, pruning may reduce the flowers for the following year.   

As the article below states, this plant is a slow grower so you can fertilize in spring to encourage a faster growth.   But it sounds like that is not what you are looking for and you shouldn't need to fertilize them just because you pruned.  I suspect next year you will see blooms again.

https://hgic.clemson.edu/factsheet/fothergilla/#:~:text=Fothergilla%20species%20are%20typically%20slow,Apply%20mulch%20to%20landscape%20beds.

Hope that helps.