Local mug served up with endless cold and hot soup (6)
I believe the answer is:
Here is my best explanation:
'soup' is the definition.
(borsch is a kind of soup)
I cannot understand how the rest of the clue works.
(Other definitions for borsch that I've seen before include "Beetroot soup", "Russian beetroot soup", "Soup made with beetroot" and "Russian or Polish soup usually containing beet juice".)