Let $C$ is a smooth, projective, geometrically irreducible curve over $\mathbb{F}_q$ and $\infty$ a closed point of $C$. A Drinfel'd module over $A = H^0(C, \mathcal{O}_C)$ is an injective homomorphism $A \rightarrow L\{\tau\} \cong \text{End}_L(\mathbb{G}_a)$, where $L$ is some characteristic $p$ field.
If $C = \mathbb{P}^1$, then $A = \mathbb{F}_q[t]$ and it is easy to write down such a homomorphism (one only needs to specify the image of $t$). However, if $C$ is a higher genus curve, then I don't see how to write down an example. Even for the case of a hyperelliptic curve, it's not clear to me how to show that there are any examples of Drinfel'd modules.
Question: how can one explicitly write down Drinfel'd modules for higher genus curves, or at least show abstractly that plenty of nontrivial examples exist?
You can find explicit constructions of Drinfeld modules on elliptic curves in David Hayes' paper On the reduction of rank-one Drinfeld modules. The simplest example (page 345 of op. cit.) is obtained by considering the elliptic curve $E/\Bbb F_2$ defined by the (affine) equation $$Y^2+Y=X^3+X+1.$$ Denote by $\infty$ the point at infinity of $E$ and by $K=\Bbb F_2(E)$ its field of rational functions. Set finally $A=H^0(E-\{\infty\},\mathcal O_E)=\Bbb F_2[X,Y]$. We can then define a Drinfeld module $\varphi:A\to K\{\tau\}$ by setting $$\begin{cases} \varphi_X=X+(X^2+X)\tau+\tau^2,\\ \,\\ \varphi_Y=Y+(Y^2+Y)\tau+X(Y^2+Y)\tau^2+\tau^3. \end{cases}$$