The most common and most likely answer to this crossword clue is the 11 letter word PARENTHESIS
I've seen this crossword clue in The Times
P
A
R
E
N
T
H
E
S
I
S
More clues leading to the result PARENTHESIS
- Reduce new academic work that immediately follows this
- Take care of male relative as an afterthought, perhaps?
- Digression from set phrase in translation
- Extra words from mum or dad having explosive little girl
- Badly written phrase inset that’s in brackets
- Mother, perhaps, almost completed dissertation in interlude
Buddy explains!

The straight part
We know that "as an afterthought, perhaps" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case as an afterthought, perhaps = parenthesis.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 PARENTHESIS mean?
PARENTHESIS noun- A clause, phrase or word which is inserted (usually for explanation or amplification) into a passage which is already grammatically complete, and usually marked off with brackets, commas or dashes.
- Either of a pair of brackets, especially round brackets, ( and ) (used to enclose parenthetical material in a text).
- A digression; the use of such digressions.
- Such brackets as used to clarify expressions by grouping those terms affected by a common operator, or to enclose the components of a vector or the elements of a matrix.