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Reading on Infinitary languages I'm only seeing first order infinitary languages $\mathcal L_{\kappa, \lambda}$, i.e. in all of these languages no quantification over predicate and function symbols is allowed.

Are there second (or even higher) order infinitary languages? I mean something like $\mathcal L^2_{\kappa, \lambda}$ (more generally $\mathcal L^n_{\kappa, \lambda}$)rendering of the infinitary first order language $\mathcal L_{\kappa, \lambda}$. Or, is it the case that all of those are reducible to first order infinitary languages, and so dispense with all of them?

If there are, what are the recommended sources on those?

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Sure there are. They even come up in practice from time to time; e.g. to show that assuming Vopenka's Principle the modal analogue of second-order logic (a la Hamkins/Woloszyn) has definable-in-$V$ semantics, the only argument I'm aware of goes through $\mathcal{L}_{\theta,\theta}^2$ where $\theta$ is the smallest limit of extendible cardinals. (This is basically a Los-Tarski-type argument, see here.)

That said, even finitary second-order logic is incredibly complicated. In particular, it doesn't have any good metatheorems (compare forcing absoluteness for $\mathcal{L}_{\infty,\omega}$ or complicated analogues of Barwise compactness for $\mathcal{L}_{\infty,\omega_1}$). So I'm not aware of any source treating infinitary extensions of $\mathsf{SOL}$ or higher-order logics in detail; it's more something that is treated as it shows up.

It's also worth noting that there's a finitary second-order sentence not equivalent even to any $\mathcal{L}_{\infty,\infty}$ sentence:

"The cardinality of the universe is a successor cardinal."

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