Proof that this is not a secure pseudorandom function?
Hint: multiplicative property.

What you have tried does not work because $y_1=y_2$ does not hold in general when $x_2=y_1+p$ (further, adding $p$ is identity in $Z_p$ ).

You want to build a distinguisher for $F_k$. Arguably, $F_k(1)=1$ is enough for that, but you can build a more general distinguisher from $F_k(x_1\cdot x_2\bmod p)=F_k(x_1)\cdot F_k(x_2)\bmod p$.