Which numbers can be written as the sum of an odd and an even square, and in how many ways?

Well, for one 25 meets your conditions:25 =3^2 + 4^2 =9 + 16 =25.  Another number is 85, because you can write it as: 6^2 + 7^2 =85 =2^2 + 9^2 =4 + 81 =85....and so on. There are probably an infinite number of such numbers.

The number must be odd

If it is a prime number modulo 4 = 1, it works