The most common and most likely answer to this crossword clue is the 7 letter word CHARGED
I've seen this crossword clue in The Guardian
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More clues leading to the result CHARGED
Buddy explains!

The straight part
We know that "Given responsibility" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case given responsibility = charged.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 CHARGED mean?
CHARGED verb- To assign a duty or responsibility to
- To assign (a debit) to an account
- To pay on account, as by using a credit card
- To require payment (of) (a price or fee, for goods, services, etc.)
- (possibly archaic) to sell at a given price.
- To formally accuse (a person) of a crime.
- To impute or ascribe
- To call to account; to challenge
- To place a burden or load on or in
- To load equipment with material required for its use, as a firearm with powder, a fire hose with water, a chemical reactor with raw materials
- To move forward quickly and forcefully, particularly in combat and/or on horseback
- (of a hunting dog) to lie on the belly and be still (A command given by a hunter to a dog)