As well we know when the vacuum polarization in the quantum electrodynamics emerges, the non-linear interaction between the electromagnetic fields should be considered. And the corresponding field of non-linear electrodynamics could have important effects on black hole physics. In this work, we focus on the relationship between the observable quantity--the shadow radius and the first-order phase transition for the non-linear charged AdS black hole in the frame of the Einstein-power-Yang-Mills gravity. The results show that under a certain condition there exist the first-order phase transition both from the viewpoint of the shadow radius and the horizon radius which are as the function with temperature (or pressure). From the viewpoint of the shadow radius, the phase transition temperature is higher than that from the viewpoint of the horizon radius with the same condition. This may be due to the non-linear Yang Mills charge and the gravitational effect. These indicate that the shadow radius can be regarded a probe to reveal the thermodynamic phase transition information of black holes. When the system is undergoing the phase transition in two cases of the different non-linear YM charge parameter: $\gamma=1,~1.5$, the thermal profiles of the coexistent big and small black hole phases with the temperature are presented. Furthermore, the effects of non-linear YM charge parameter on the shadow radius and the thermal profile are also investigated.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd