Ok, so my brief understanding is that an accelerating charge will produce a magnetic field. If the charged particle, say an electron is being accelerated by a magnetic field, how does the electron’s resultant magnetic field affect the original magnetic field?
According to standard theory based on Maxwell's equations in vacuum, the fields can be distinguished as independent contributions to total field.
If the electron's magnetic field doesn't influence the source of the external magnetic field, it doesn't affect this external magnetic field. This is appropriate for external magnetic field that is due to very distant source (such as radio wave source in another galaxy). The two fields can be thought of as independent contributions to total magnetic field everywhere except the point where the electron is (this assumes the classical point-like model of the electron). In other words, the electron's magnetic field doesn't change the external magnetic field.
If the electron's magnetic or electric field influences the source, such as in most situations where the field is due to nearby source such as permanent magnet or electromagnet, then the influence on the external magnetic is taking plac but it is not easy to calculate. Often one assumes the influence is negligible.
In speculative nonlinear modifications of EM theory, the fields cannot be easily distinguished by their source, some arbitrary rule has to be introduced to define what is field of the external source and what is field of the electron. Then one would see that in such theories, in a sense, one magnetic field could influence the other. However, the very concept of separate fields loses its usefulness in such theory due to arbitrariness of the attribution, and only the concept of total field remains unique.