= Solution
The identified involution is $z=g^2=h^5$. It is central: it commutes with $g$ because it is a power of $g$, and with $h$ because it is a power of $h$. We use the <normal form theorem for an amalgamated free product>: the two factors embed, an alternating product of elements outside the amalgamated subgroup is nonidentity, and fixed coset representatives give a unique reduced expression. Here choose representatives $1,g$ in the first <cyclic group> and $1,h,h^2,h^3,h^4$ in the second.
An effective expression from the <normal form theorem for an amalgamated free product> is
$$
z^\varepsilon x_1\cdots x_r,\qquad\varepsilon\in\{0,1\},
$$
where each $x_j$ is a nonidentity representative and successive representatives come from different factors. Reduce powers by $g^u=z^{\lfloor u/2\rfloor}g^{u\bmod2}$ and $h^v=z^{\lfloor v/5\rfloor}h^{v\bmod5}$, using these identities for negative integers too. Move the central $z$ factors left and reduce their total exponent modulo two. Whenever adjacent representatives come from one factor, multiply them, split off their $z$ contribution, and delete a resulting identity. Each such combination reduces the number of representatives, so the procedure terminates. By the <normal form theorem for an amalgamated free product>, \b[the input is trivial exactly when $\boxed{r=0\text{ and }\varepsilon=0}$.] In particular $z\ne1$ because each factor embeds. This gives the <soluble word problem in finite cyclic amalgams> algorithm without merely asserting that a normal form exists.
In the specified quotient, the extra <relator> makes $g,h$ commute. Put $t=gh^{-2}$. Its abelian <group presentation> gives
$$
t^5=g^5h^{-10}=g,\qquad t^2=g^2h^{-4}=h,\qquad t^{20}=1.
$$
Conversely $g\mapsto s^5$, $h\mapsto s^2$ satisfy every quotient relator in $C_{20}=\langle s\mid s^{20}=1\rangle$, and send $t$ to $s$. Thus the quotient is exactly $C_{20}$, rather than just a cyclic group of order dividing twenty. If $M,N$ are the exponent sums of $g,h$ in the input, \b[its quotient word problem is]
$$
\boxed{w=1\quad\Longleftrightarrow\quad5M+2N\equiv0\pmod{20}.}
$$
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