The question's observation is perfectly right: RSA as practiced uses padding or/and is used to encipher a symmetric key, which makes it not homomorphic at all.
Further, textbook RSA (without padding) is homomorphic, but is deterministic (the same plaintext always produces the same cypertext). This is most often a devastating weakness, because it lets anyone (with the public key) test an hypothesis on the plaintext.
For this reason, use of RSA as an homomorphic asymmetric encryption algorithm is questionable, and accordingly is now almost entirely absent in cryptographic publications with a peer-review system that I trust.
Update: A related question asks if it is possible to devise safe RSA padding while preserving homomorphy; I tentatively answered in the affirmative, but that does not seem practical at all.
I do not know any way to make RSA fully homomorphic, in the usual sense that it allows both addition and multiplication of plaintext from ciphertext.
The Pailler cryptosystem is a randomized asymmetric encryption algorithm (safe from plaintext guessing); is homomorphic (for addition of plaintext, rather than multiplication for RSA); is based on the same mathematical problem as RSA; and is not much more difficult to grasp. It might be a proper substitute for RSA when a homomorphic asymmetric encryption algorithm is thought. Also, there seems to be more applications of additive homomorphic encryption than of the multiplicative breed.
Among applications of the Pailler cryptosystem (and homomorphic encryption in general) are some electronic voting protocols (an interesting theoretical subject; still, count me among the strong opponents of their actual use for political elections).