He was entrusted with something and did not take proper care of it, and he has no wealth; could he be given zakaah to pay off this trust?

Firstly:

Allah, may He be exalted, says (interpretation of the
meaning):

“Indeed, Allah commands you to
render trusts to whom they are due”

[an-Nisa’ 4:58].

What this person should have done is render back the trust
with which he had been entrusted, and he had no right to spend it on
whatever need arose.

Secondly:

Allah, may He be exalted, says (interpretation of the
meaning):

“Zakah expenditures are only for
the poor and for the needy and for those employed to collect [zakah] and for
bringing hearts together [for Islam] and for freeing captives [or slaves]
and for those in debt and for the cause of Allah and for the [stranded]
traveler – an obligation [imposed] by Allah . And Allah is Knowing and Wise”

[at-Tawbah 9:60].

What appears to be the case is that this man comes under the
ruling on one who borrowed money for sinful purposes. The basic principle
concerning that is that if he repents and regrets betraying the trust, but
he has no money surplus to his needs with which to pay off the debt that he
still owes, then in this case he is a debtor to whom it is permissible to
give zakaah wealth with which to pay off his debt.

But if he is persisting in sin, and there is no sign on his
part to indicate that he has repented, then he should not be given zakaah.

The same applies to one who did that because he was unaware
of the prohibition on his doing that, and the prohibition on taking anything
from what had been entrusted to him, even if he had the intention to borrow
it and pay it back, without the permission of the rightful owner.

And the same applies to one who did that on the basis of
wrong thinking, or thinking that this was permissible for him, so long as
his intention was to pay it back, as some of the common folk do in such
cases.

In that case, there is nothing wrong with giving him zakaah
and paying off his debt from the zakaah of one’s wealth, if he has repented,
or he did that out of ignorance, or on the basis of mistaken thinking.

It is preferable to give the money to this person, so that he
can give it himself to the one to whom he owes the debt.

But if there is the fear that he may spend it on his own
needs, because he is used to being extravagant and does not handle money
responsibly, then in this case it is permissible to give the money directly
to the person to whom the debt is owed, with the intention of paying off the
debt on behalf of this person.

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

Is it permissible for us to go to the lender and give him his
money without the knowledge of the debtor?

Answer: Yes, that is permissible, because it is included in
the words of Allah, may He be exalted, “and
for freeing captives [or slaves]”
[at-Tawbah 9:60]; this is followed by the phrase
“and for those in debt”,
which is mentioned in conjunction with [and with similar grammatical
structure to] the preceding phrase. The conjunction indicates that it does
not mean that it is essential to give it to the debtor, so it is permissible
to give it directly to the lender (on behalf of the debtor).

If someone were to ask: Is it better for us to give the money
to the debtor, so that he can give it to the lender, or to give it to the
lender ourselves? The answer is that it depends:

If the debtor is trustworthy and keen to pay off his debt,
then it is undoubtedly more appropriate to give it to him so that he can pay
off the debt himself, and will no longer be embarrassed or be criticised by
people.

But if there is the fear that he may mishandle this money,
then we should not give it to him; rather we should go to the lender who is
asking him to pay off his debt.

Question: if someone borrowed money for unlawful purposes,
can we give him zakaah funds (to pay off his debt)?

Answer: If he has repented, then we may give it to him,
otherwise we should not give it to him, because that is helping in something
unlawful, and if we give it to him he would borrow money for the same
purpose again.

End quote from ash-Sharh al-Mumti‘ (6/234-235)

And Allah knows best.