Is this zero knowledge protocol for honest verifiers? Assume the zero knowledge protocol where the prover knows a $x$ such that: $g^x = h \pmod{p}$.

The prover chooses a random $t \in \mathbb{Z}^*_m$ and sends $y = g^t \pmod{p}$
The verifier sends random $c \in \mathbb{Z}^*_m$ and sends it
The prover calculates $s = t + c + x$ and sends it
The verifier accepts if and only if $g^s = yg^ch \pmod{p}$

Is this for honest verifiers?
Attempt:
The real transcript is $(t, g^t \pmod{p}, c, t +c+x)$, but I cannot find a simulated transcript $(t, \cdot, c, \cdot)$ that has the the same distribution as the real one. Can you give me some hints?
Can you give me some hints?

It's not a proof of knowledge, as someone without knowledge of $x$ can complete this protocol successfully with an honest verifier.
Hint: what happens if the ignorant prover sends $y = h^{-1}$?