There are many sources of the relations and Hopf algebra structure of quantum affine $sl_2$ as a deformed enveloping algebra. However, for an application to integrable systems I need to look at quantum affine $gl_2$, not $sl_2$. Also it would be really good to see how this fits into the Drinfeld realisation with the $h_r$ and $x^\pm_r$ for $r\in\mathbb{Z}$. I would be very grateful if someone could point me in the right direction.
I guess you mean the following presentation in terms of generators and relations:
The excerpt is from:
 Evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras, arXiv:1709.01592v4 [math.QA], p.3
I think the OP might also find some interest at:
 On Drinfeld realization of quantum affine algebras, Naihuan Jing, arXiv:qalg/9610035
 MULTIPLICATON FORMULAS AND CANONICAL BASES FOR QUANTUM AFFINE ${\mathfrak{gl}}_n$
You can also see (both Jimbo's and Drinfeld's presentation of the quantum affine $sl_2$) at:

$\begingroup$ Many thanks for that! Is there any particular reason why the degree operator is left out? I assume that it could be put back in... $\endgroup$ May 20 at 7:31