Is this a type of fallacy?

I worked in an HR Department for a large company, where I would sometimes get requests from employees for pay advances, which were allowed but within set limits, regardless of the circumstances. And yet some employees would still try to exceed those limits, and when I had to reluctantly turn down the request, they often countered with, "In other words, you're happy to see my family out on the streets/going hungry/without Christmas presents etc..." None of which would make me happy.

Is this an example of a particular type of fallacy?

Yes, this is an example of a particular type of fallacy known as an **appeal to emotion (argumentum ad passiones)**. This fallacy occurs when someone tries to manipulate an emotional response in place of a valid or compelling argument. In this instance, the employees are attempting to evoke feelings of guilt or compassion by painting a dire picture of the consequences of not granting their request for a pay advance, rather than providing a logical reason why the policy should be bypassed in their specific case.

Key elements indicating this fallacy:
- **Evoking Emotional Reaction:** The employees’ statements about their families being out on the streets, going hungry, or without Christmas presents are designed to provoke an emotional reaction.
- **Absence of Logical Justification:** Instead of providing a rational argument for why their request should be granted beyond the set limits, they rely on eliciting sympathy or guilt.

By focusing on the emotional impact rather than presenting evidence or reasoning that aligns with company policy, they attempt to sway the decision-making process in their favor.

To what AI Fallacy Master said, add the Fallacy of the False Alternative (also known as the false dilemma, false dichotomy, fallacy of bifurcation, and black-or-white fallacy) is when a proponent asserts two alternative statements are the only possible options, but there are more options

And, the Fallacy of Diversion is when one party attempts to divert the discussion to something else, in this case, your feelings. 

There are others, but this will get you started.