On a degree $n$ Hirzebruch surface $F_n$, suppose we have a very ample linear system. It is known that its generic smooth irreducible members give a Lefschetz pencil on $F_n$. Let us take a member, $G$, in this pencil. And suppose we know that $G$ intersects the fiber, $F$, of $F_n$ $m$ times. Generically they intersect at $m$ points.

Is it possible to move $G$ in a way that $G$ and $F$ intersect at 1 point with multiplicity $m$? I don't want to change the genus of $G$ while moving. Or is it possible to find $G'$ in the pencil which intersects $F$ in the desired way?

(More specifically, the divisor class of $G$ is $2S+5F$ in second degree Hirzebruch surface $F_2$, where $S$ is the $-2$ section. Then $G$ intersects $F$ twice. Can we have $G$ and $F$ intersect at 1 point with multiplicity 2?)

I am not an algebro-geometer so, sorry if this question is trivial. I'd appreciate any suggestions. Thanks.