The most common and most likely answer to this crossword clue is the 4 letter word SHAG
I've seen this crossword clue in Independent
S
H
A
G
Buddy explains!

The straight part
We know that "with second ugly old woman (4)" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with second ugly old woman (4) = shag.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 SHAG mean?
SHAG noun- Matted material; rough massed hair, fibres etc.
- Coarse shredded tobacco.
- A type of rough carpet pile.
- Bacon or fat, especially if with some remaining hair or bristles.
- A roughly-cut or torn-off piece of bread or cheese.
- To make hairy or shaggy; to roughen.
- To hang in shaggy clusters.
- Hairy; shaggy.
- Several species of sea birds in the family Phalacrocoracidae (cormorant family), especially the common shag or European shag, Phalacrocorax aristotelis, found on European and African coasts.
- A swing dance.
- An act of sexual intercourse.
- A casual sexual partner.
- To shake, wiggle around.
- To have sexual intercourse with.
- To have sexual intercourse.
- To masturbate.
- To chase after; especially, to chase after and return (a ball) hit usually out of play.
- To perform the dance called the shag.
- (Northwestern Ontario) A fundraising dance in honour of a couple engaged to be married.
- Friend; mate; buddy.