The most common and most likely answer to this crossword clue is the 5 letter word BORNE
I've seen this crossword clue in The Guardian
B
O
R
N
E
Buddy explains!

The straight part
We know that "Put up with" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case put up with = borne.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BORNE mean?
BORNE verb- To carry or convey, literally or figuratively.
- To support, sustain, or endure.
- To support, keep up, or maintain.
- To press or impinge upon.
- To produce, yield, give birth to.
- (originally nautical) To be, or head, in a specific direction or azimuth (from somewhere).
- To gain or win.
- Carried, supported.