The most common and most likely answer to this crossword clue is the 4 letter word BEAR
B
E
A
R
More clues leading to the result BEAR
Buddy explains!

The straight part
We know that "Put up with/Paddington, for example" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case put up with/paddington, for example = bear.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 BEAR mean?
BEAR noun- A large omnivorous mammal, related to the dog and raccoon, having shaggy hair, a very small tail, and flat feet; a member of family Ursidae.
- A rough, unmannerly, uncouth person.
- An investor who sells commodities, securities or futures in anticipation of a fall in prices.
- A state policeman (short for smokey bear).
- A large, hairy man, especially one who is homosexual.
- A portable punching machine.
- A block covered with coarse matting, used to scour the deck.
- The fifteenth Lenormand card.
- Something difficult or tiresome; a burden or chore.
- To endeavour to depress the price of, or prices in.
- (investments) Characterized by declining prices in securities markets or by belief that the prices will fall.
- 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.
- Barley, especially four-rowed barley or six-rowed barley.
- A pillowcase; a fabric case or covering as for a pillow.