Can I use a professor's letter of reference as a recommendation?
Three things:

1) It's polite to ask someone before putting their name forward as a reference. It's a huge turnoff to have someone get in touch with you unexpectedly saying that you're a reference for someone you didn't think you were a reference for.

2) It's smart to ask someone before putting them forward as a reference, because it gives them a chance to respond to you. It's possible that this person may not be able to give you a good reference at this point in time, and nothing will destroy your chances faster than if the admissions committee follows up with a reference who then says they would not recommend you.

3) In this case it sounds like the letter of reference might be highly specific to that institution: "I have the funding to work with this student and want them to do a PhD with me." That's a nice thing to say about someone, but it's not really a general purpose reference. Ideally, applying to a PhD program, you want a letter to explain why you're a great asset to a research program.

If you still want to use this letter, then be sure to ask the reference before you do so.