TL;DR: A husband may refuse khul' regardless of whether or not the wife is capable of repaying the mahr. Among the reasons a husband may refuse khul' is the inability of the wife to pay the husband an agreeable compensation. Ordinarily, khul' requires returning an amount equivalent to the mahr, but the husband can agree to less.
It seems that the mahr is normally the amount to be returned.
Khul' is when a wife demands divorce if she does not like her husband for any reason. She can do so giving him back the bridal money he gave her.
This appears to be due to the hadith Sunan an-Nasa'i 3463 [grade: sahih]. However, there are fatawa that say that this is not a strict requirement:
It is up to the man and his wife to agree on a lesser compensation. Most scholars agree that the compensation may be more than the mahr itself, but some scholars say that this is not permissible.
Dar Al-Ifta (see also Islam Q&A)
The Dar Al-Ifta fatwa continues
A man who does not have anything to give as mahr must not be offended if his marriage proposals are rejected. Similarly, a woman who cannot compensate her husband for what he stands to lose as a result of the termination of marriage is not to feel aggrieved if he refuses. But in these matters, it is recommended to show forbearance and kindness to others. Even though we may be unhappy with the termination of the marriage, as a community we should look into helping her.
This indicates that the husband may refuse the divorce if she cannot afford it. She may then take it to a sharia court; see another Dar Al-Ifta fatwa for the technical details.
A Darul Iftaa fatwa describes the case that the husband is at fault:
However, the Fuqaha state that, if the husband was at fault and it was his wrongdoings that resulted in the failure of their marriage, then it is impermissible for him to demand a financial payment in return for a divorce.