Well, I'm going to disagree with just about everybody else. I would say yes, go ahead and help, but the way you do it is key.
In your shoes I would tell the woman you're working with that someone who wants to remain anonymous has suggested that she would like to help out a client by paying a bill and that you would like to nominate this lady. Then you can make arrangements to pay the bill without handing over any cash at all. I do this frequently in a number of situations and all you need are the recipient company's bank details and a name, address and reference number for the account being paid.
Doing it this way means that you are not setting a precedent for handing over cash, you know the bill will be paid and you can sleep easy knowing that you have alleviated at least the current crisis.
In my role I am in a position to make conditions when I'm administering money and I would strongly suggest to the client that she engages with CAP (Christians Against Poverty) or Stepchange in order to ensure that her financial difficulties are addressed.
What is the most random school memory you have?
and this lady is lucky to know you .