The most common and most likely answer to this crossword clue is the 4 letter word HIKE
I've seen this crossword clue in Independent
H
I
K
E
Buddy explains!

The straight part
We know that "Put up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case put up = hike.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 HIKE mean?
HIKE noun- A long walk.
- An abrupt increase.
- The snap of the ball to start a play.
- A sharp upward tug to raise something.
- To take a long walk for pleasure or exercise.
- To unfairly or suddenly raise a price.
- To snap the ball to start a play.
- To lean out to the windward side of a sailboat in order to counterbalance the effects of the wind on the sails.
- To pull up or tug upwards sharply.