Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a Dehn twist $\tau \in G$ which is not in $H$ but such that $\tau^k$ is in $H$, $k$ being minimal for this property ?

Using the chain relation, one can construct examples with $k = 2$, but I cannot find any examples with higher powers.

Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a Dehn twist $\tau \in G$ which is not in $H$ but such that $\tau^k$ is in $H$, $k$ being minimal for this property ?

Using the chain relation, one can construct examples with $k = 2$, but I cannot find any examples with higher powers.

EDIT: I realized I should put a little motivation to this question. So let us take $U_d$ the space of smooth degree $d$ planar complex curves. There is a monodromy map $\pi_1(U_d)\rightarrow G$, and the image of this map is generated by Dehn twists obtained for example by taking a generic pencil of curves. Now I can construct a loop in $U_d$ whose monodromy is a power of some Dehn twist and I am wondering if there should be a reason coming from the structure of $G$ for the Dehn twist itself to be in the image of the monodromy.