Sushi and Tempura were the only major requests from visitors I met 5 years ago. In recent years requests are much more varied, from ramen noodles to Kobe beef. Some would like to try a bento box others want Japanese curry. I feel Japanese food has become much better known overseas.
Kyoto's well-known cooking styles are;
Obanzai (the home cooking Kyoto style)
Kaiseki cuisine (both Kaiseki as course meal 会席 and meal at tea ceremony 懐石)
Shojin cuisine (Buddhist vegetarian meal).
There are many headquarters of temples in Kyoto and they all receive visitors from the parishioners of their branch temples. Some of them offer accommodation and will provide food or have catering services. I mentioned before in this blog about Okutan (a yudofu restaurant) which is one of this kind of facility at Nanzen-ji Temple. Okutan is a private restaurant, not run by the temple.