Assume a finite commutative base group $\mathbb B$, some $g$ in $\mathbb B$, and $\mathbb{G} = \langle g\rangle$ the subgroup that $g$ generates, with choice of $\mathbb B$ and $g$ such that ElGamal encryption would be secure for a random message in $\mathbb G$ (semantically/under CPA).
ElGamal encryption still allows decryption if we use a random message in $\mathbb B$ rather than in $\mathbb G$. How much information about the message can leak if we do this? If necessary to get a result, restrict to $\mathbb B=\mathbb Z^*_p$, perhaps with $p$ an odd prime, and/or to $\mathbb G$ of prime order.
Notation: order of $\mathbb G$ generated by $g\in \mathbb B$ (noted multiplicatively) is $q$; private key $x$ is uniformly random with $0<x<q$; public key is $h=g^x$. Encryption of $m$ random in the message space ($\mathbb G$ for the standard definition of ElGamal encryption, $\mathbb B$ in the question) generates one-time $y$ uniformly random with $0<y<q$, and computes $c_1=g^y$, $s=h^y=g^{x\,y}$, $c_2=m\,s$. Ciphertext is $(c_1,c_2)$. Decryption recomputes $m$ as $c_1^{q-x}\,c_2$.
It is known that if $\mathbb B=\mathbb Z^*_p$ with $p$ a large prime and $\mathbb G$ of prime order $(p-1)/2$, there's an information leak of one bit: we can determine from the ciphertext if the plaintext is a quadratic residue or not; but nothing else as far as we know. I'm unsure about the many other cases ($\mathbb G$ of smaller prime order, or not of prime order; other base groups).
My motivation for that problem came while answering this question.