The most common and most likely answer to this crossword clue is the 11 letter word SUPPOSITORY
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Buddy explains!
The straight part
We know that "medical aid via back door" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case medical aid via back door = suppository.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 SUPPOSITORY mean?
SUPPOSITORY noun- A medicine in the form of a small plug that is inserted into a bodily cavity, especially the rectum, vagina or urethra, where it melts at body temperature.