The most common and most likely answer to this crossword clue is the 2 letter word PI
I've seen this crossword clue in The Times
P
I
Buddy explains!

The straight part
We know that "Vagrant mongrel" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case vagrant mongrel = pi.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 PI mean?
PI noun- The 16th letter of the Classical and Modern Greek alphabets and the seventeenth in Old Greek.
- An irrational and transcendental constant representing the ratio of the circumference of a Euclidean circle to its diameter; approximately 3.14159265358979323846264338327950; usually written π.
- Metal type that has been spilled, mixed together, or disordered.
- To spill or mix printing type.
- Not part of the usual font character set; especially, non-Roman type or symbols as opposed to standard alphanumeric Roman type.
- Pica (conventionally, 12 points = 1 pica, 6 picas ≈ 1 inch)
- Piaster
- Pious