Hi Anissa,
How about:
Imperial Palace:
Japanese Beef (Wagyu) near the Imperial Palace (lunch or dinner):
Hafuu: located on Fuyacho Street just south of the Imperial Palace. Closed Wed. While wagyu is never very cheap, Hafuu offers high quality and good cost performance. Lunch is of course cheaper than dinner. I think that the Hafuu Katsu Sandwich is available take out and is REALLY excellent!
http://www.hafuu.com/honten/index.html
Souvenirs near Imperial Palace:
Handcrafted bags from Cream (not especially cheap) Scroll down to the bottom of the article for Cream.
https://openkyoto.com/kyoto-support/topic/kyoto-souvenir-original-handbag-tote-bags-purses-wallets-and-rucksack-makers
In Gion
The weather is getting hot and Gion Koshi serves excellent parfaits and chilled sweet treats, not much cheaper yummies in Gion.
KyotoFoodie article: http://kyotofoodie.com/gion-koishi-kakigori-summer-time-treat-in-gion/
The original ‘Western’ food in Japan: (very cheap)
http://kyotofoodie.com/issen-yoshoku-the-origins-of-western-food-in-japan/
Gion noodle shop frequented by Kabuki actors and Maiko/Geisha famous for it’s noodle dishes with heaps of Kyoto scallions.
http://kyotofoodie.com/negi-udon-shop-yorozuya-gion-kyoto/
Foodie Souvenirs in Gion:
Traditional Spice Store Hararyokaku
https://openkyoto.com/kyoto-support/topic/where-to-purchase-condiments-sauces-and-spices-in-kyoto
I will try to think of a few more. How is this list so far?