The most common and most likely answer to this crossword clue is the 4 letter word BEGS
B
E
G
S
Buddy explains!
The straight part
We know that "Starts to beseech, entreat, grovel and sponge" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case starts to beseech, entreat, grovel and sponge = begs.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 BEGS mean?
BEGS noun- The act of begging; an imploring request.
- To request the help of someone, often in the form of money.
- To plead with someone for help, a favor, etc.; to entreat.
- In the phrase beg the question: to assume.
- In the phrase beg the question: to raise (a question).
- To ask to be appointed guardian for, or to ask to have a guardian appointed for.
- A provincial governor under the Ottoman Empire; a bey.