Is this a structural weakness of Feistel networks? I'm doing a lot of reading about Feistel networks. Something occurred to me a bit ago that I hadn't realized previously, namely that in any Feistel construction there are bits of the plaintext that are never given as input to the round function without having been mixed with the previous round function's output. 

For example, in the first round of a balanced Feistel network $L_0$ is XORed with $F_k(R_0)$ to get $R_1$. Thus, $L_0$ is never used by itself as input to $F_k$, which could violate the diffusion property of the cipher. 

Is it even possible to get around this limitation, perhaps with a modified first round?

I don't think this is a real vulnerability in e.g. DES or Blowfish, for the record. Rather, I'm wondering whether this could be used to craft a successful attacker in a game against a reduced-round variant of these kinds of ciphers.
If such a network had only a single round, then you might have a valid concern. This is why there needs to be least three rounds, so that every bit from L can potentially affect every other bit from L (via R from the second round).

It isn't a structural flaw, because multiple rounds are assumed. Changing this round structure would mean that it was no longer a Feistel network, and you would lose the reversibility of each round (when the key is known).