Waring number for a word

In the group sense
Suppose $$G$$ is a group, $$w$$ is a defining ingredient::word in $$n$$ letters, and $$H$$ is the verbal subgroup of $$G$$ generated by the image of the word map $$w$$ in $$G$$, i.e., $$H$$ has as a generating set the set:

$$A = \{ w(g_1,g_2,\dots,g_n) \mid (g_1,g_2,\dots,g_n) \in G^n \}$$

The Waring number in the group sense for $$w$$ is defined as the diameter for the generating set $$A$$ of $$H$$, i.e., it is the smallest $$k$$ such that every element of $$H$$ can be expressed as a product of length at most $$k$$ involving elements of $$A$$ and their inverses.

In the group sense
Suppose $$G$$ is a group, $$w$$ is a word in $$n$$ letters, and $$H$$ is the verbal subgroup of $$G$$ generated by the image of the word map $$w$$ in $$G$$, i.e., $$H$$ has as a generating set the set:

$$A = \{ w(g_1,g_2,\dots,g_n) \mid (g_1,g_2,\dots,g_n) \in G^n \}$$

The Waring number in the monoid sense for $$w$$ is defined as the smallest $$k$$ such that every element of $$H$$ can be expressed as a product of length at most $$k$$ involving elements of $$A$$. The key difference with the previous definition is that we do not include inverses. Note that if $$A$$ is a symmetric subset (as is the case with the images of most word maps of interest to us) then these two Waring numbers are the same).