I agree. "Only" should come AFTER "have" to modify "sixty cents." It is misplaced to put it before "have" because then it modifies "have" and the meaning becomes you just HAVE the money and don't do anything else with it. (Though colloquial this arrangement is often used the same way, even though it is not correct.)
RyagonIV's distinguishing between having only a certain amount and having that amount but expecting more by shifting the position of "only" is interesting, but I suspect doesn't work for most English-speakers. At least for most Americans, "I only have X" is the colloquial way of saying "I have only X." They mean the same thing.
Taken literally, "I only have sixty cents" is never true. The person has other things as well, say, bluejeans, a T-shirt, a driver's license, etc. But no one takes the statement literally. Whereas, "I have only sixty cents" can be true, if you think of "only sixty" as modifying "cents."
I too believe that both "cent" and "centavo" should be accepted, since we don't know the context of the statement. Now, if the statement contained "dollars," like, "Tengo cinco dolares y ocho cantavos," then "cents" should be the appropriate answer. Otherwise, as I say, either one should be accepted, especially since those studying Spanish are probably intending to use it in a place where pesos and centavos are the medium of exchange.