A Magician-Esper Hybrid is a person who can use magic and esper abilities. However, since magic and esper abilities have completely different and incompatible formats, and the fact that espers' brains have been modified due to the Power Curriculum Program, the hybrid will suffer severe damage from performing magic.
The only "successful" hybrid is Tsuchimikado Motoharu, as a result of him joining the Power Curriculum Program in order to complete his ruse of being an ordinary student, him being an operative sent to spy on the city. However, this is only due to his esper ability being able to repair any damage inflicted to him, which could fail at any time. Because of his hybridization, Motoharu, a talented magician in his own right, can only perform magic a limited number of times before being grievously injured.
There was an experiment to create magician-esper hybrids by Academy City which involved sending espers to learn magic, which Sherry Cromwell was a part of. This experiment was a failure not only due to the injuries the subjects sustained from using magic, but the Knights of England killed off the subjects, including Ellis, Sherry's friend, due to the belief that the experiment was against the truce between the magic and science sides.
There is currently no way for a magician or esper to become a hybrid without the dangers present with the incompatibility of the formats between magic and esper abilities.
Kuroyoru Umidori, the cyborg esper had once abused the negative side effect of an esper using magic to create an avalanche. Being a cyborg, what counted as her body was vague, by imagining something to be part of her, for example such as a cliff, she can force the negative side effect on a target by using any random magic spell. It is speculated that espers may use this technique in the future, to use as an attack or to mitigate the damage on one's own body when casting magic. Such techniques may prove to be very useful for Level 0's such as Hamazura Shiage and may give Motoharu more freedom to use his own magic.