sentence structure

food prices hit record high, with no relief in sight.

"with no relief" is not a clause (it does not contain a verb), but a phrase. Yes, "with" is a preposition here, so "with no relief" is a prepositional phrase, acting as an "adjective phrase", to qualify the subject "food prices".

A simple prepositional phrase contains a "preposition" and a noun, and can have more complex forms, e.g. "preposition" + "modifier" + "noun", as in "with no relief", where "no" is the modifier, and "relief" is the noun.

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With is a preposition, suggesting an accompanying situation.

The structure is "with [ no relief (in sight) ]", where "no relief" is a simple noun that is further qualified by "in sight". "in" is a preposition, and "sight" is a noun, and together they mean "in the near/foreseeable future", or "immediate."

The sentence can also be written as:
Food prices hit a record high and there is no relief in sight.