What happens to Jesus when received in a unworthy manner in the Eucharist? There is a lot of written on what happens to a person when receiving the Eucharist in an unworthy manner but not so much what happens to Jesus.

Since the whole Christ is truly present, body, blood, soul, and divinity, under the appearances of bread and wine and not just his divine nature it is ok to ask. If it's just his nature we could say "nothing much" since nature doesn't do things in itself. But since Christ is present with his soul and the body, He must experience effects of the act on the host.

Does the Catholic Church have any writings or is there any work done by some mystics or saints? This also includes when someone drops a consecrated host and small pieces are still on the ground and people step on it unknowingly.
The short answer: nothing “happens to Jesus” when a person receives the Eucharist unworthily, when the Eucharist is inadvertently stepped on, or even when it is desecrated deliberately.

As the O.P. correctly points, out, when Catholics (and Orthodox) say that Jesus is really present in the Eucharist, they affirm that he is present both in his divinity and in his humanity. Jesus is one and whole (he is the Divine Person of the Son), and he cannot be divided. (See Catechism of the Catholic Church [CCC], 1377.)

It should be clear that nothing can “happen” to Jesus as regards his divinity, because the Divine Nature cannot be changed or influenced by anything, least of all the actions of one of His creatures. (If He allows Himself to be influenced by prayers and so forth, that is a deliberate choice on His part that does not produce a change in Him of any kind; see, e.g., Hebrews 13:8, and a long Patristic tradition.)

On the other hand, Jesus in his human nature can (like any other man) undergo changes of various kinds.

As regards unworthy reception of the Eucharist and similar situations, however, there are two important considerations.

First of all, Jesus retains his human nature in Heaven and will retain it for all eternity. (See CCC 659.) However, because he has ascended into Heaven, he is now glorified and therefore impassible; that is, he can no longer suffer. (This is a property of all glorified bodies, including Christ’s; see Revelation 21:4.) Therefore, much as Jesus does not approve of unworthy reception or desecration of the Eucharist, he is not “pained” by it, at least not any longer.

(It is possible that, with his divine foreknowledge, he was aware of future unworthy receptions of the Eucharist, even as he walked this earth 2000 years ago, and that these caused him pain back then, but they do so no longer.)

Second, it should be noted that, although the presence of Jesus in the Eucharist is real and substantial (that is, the consecrated bread and wine are no longer actual bread or wine, but Jesus), it is a sacramental presence. (See CCC 1353 and 1369.) One is not to imagine that our consumption of the sacred species causes harm to Jesus physical body that is in Heaven. In a similar way, it is not as though Jesus feels a physical discomfort or anything like that when someone receives him unworthily.

It follows that no action on our own part can harm or otherwise grieve Jesus in the Eucharist (although, of course, deliberately receiving Christ unworthily or desecrating the Eucharist is extremely harmful for the person who does it).

(Naturally, people are not responsible inadvertently stepping on small particles—provided it is truly inadvertent and not due to negligence or something like that. In that case, it does not even harm or affect the persons who do it, still less Jesus.)