This is a follow-up question to my previous post: Sufficient conditions to tell whether a surface contains a line

Suppose that $f(x_1, x_2)$ is an irreducible binary form with integer coefficients of the variables $x_1, x_2$ over $\mathbb{Q}$ say, and suppose that $g(x_1, x_2)$ is another binary form with integer coefficients (not necessarily the same degree as $f$) that does not have a common component as $f$. In the case I am interested in this condition is satisfied vacuously since we assume $\deg(g) < \deg(f)$. Now, can the surface $$f(x_1, x_2) = g(x_3, x_4)$$ (in weighted projective space) contain any lines?

Note: the case $f = g = x^d + y^d$ has been treated by Heath-Brown, I believe.