Puppets only produce building, and they prioritize gold production.
If you don't want to keep puppets, annex and rushbuy a courthouse. You'll get a culture hit however (count as a founded-by-yourself city) and unhappiness. Be careful if you are building a National Wonder (such National College, Iron Works,..), get sure you finished the wonder before to annex (otherwise you'll need to buy the appropriate building to finish the wonder)
If you don't want extra unhappiness, raze the city, if the location is not interesting (not on a hill, not along a river, no luxury or strategic resources)
In vanilla you could be able to raze the city after you puppeted it, but you'll get the culture malus.
I always puppet for domination, I annex only the capitols and city over 10 citizens, I check also inf the city still has happiness building (coliseums, etc)
If your cities are building temples and librairies, well, that's the deal. Or pick Rome to get the 25% production bonus for all building already built in capitol, or pick the liberty SP (+5% production in cities)
I suggest you to buy G&K and BNW as soon as possible!