Reduce $SAT$ to $3SAT$
Let's take the case $k = 1$. Consider the CNF $z_1 \land \lnot z_1$. You suggest replacing it by
$$
(z_1 \lor y_{1,1} \lor y_{1,2}) \land (\lnot z_1 \lor y_{2,1} \lor y_{2,2}).
$$
Hopefully you see the problem now.