The most common and most likely answer to this crossword clue is the 5 letter word FRUMP
I've seen this crossword clue in The Daily Telegraph
F
R
U
M
P
Buddy explains!

The straight part
We know that "Dowdy woman" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dowdy woman = frump.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 FRUMP mean?
FRUMP noun- A frumpy person, somebody who is unattractive, drab or dowdy.
- The clothes that such a person would wear.
- A bad-tempered person.
- A flout or snub.
- To insult; to flout; to mock; to snub.