No, it doesn't go against the idea of specific heat. You may be thinking of ideal gases and even then specific heat may or may not be constant: https://en.wikipedia.org/wiki/Perfect_gas
The ideal gas model assumes point particles with no interparticle interactions, while all the interesting stuff with regards to all kinds of specific heat happens due to these things in... particular.
> Why would the amount of energy required to increase the temperature of a defined mass of water depend on the temperature?
and the answer is that it does, except if you have a very restrictive model where it doesn't. (Edit: I see there was an edit to the previous comment, I had missed that.)
It doesn't. Your initial assumption that the heat capacity is a thing that doesn't depend on temperature did (because it may apply to ideal gases as opposed to water where it just doesn't stay constant as you noticed later).
What does that have to do with a perfect gas?