There is a newly constructed Cafe next to the Albanian Church in Kish. They can be contacted at +944506310246. The owner of the cafe also administers the Albanian Church, so it is possible to speak a little English with her. They serve tea with a plethora of jam if you just want tea though, it will cost 1 AZN. In addition to tea, they have soup and kebab.
It is possible to eat at one of the tea houses in Kish. If you are a woman, make sure to ask for a private room in order to keep from disturbing local customs.