In this note we explore a non-static spacetime in quantum regime in the background of $f(R)$ gravity. The time dependent Vaidya metric which represents the spacetime of a radiating body like star is studied in an energy dependent gravity's rainbow, which is a UV completion of General Relativity... In our quest we have used gravitational collapse as the main tool. The focus is to probe the nature of singularity (black hole or naked singularity) formed out of the collapsing procedure. This is achieved via a geodesic study. For our investigation we have considered two different models of $f(R)$ gravity, namely the inflationary Starobinsky's model and the power law model. Our study reveals the fact that naked singularity is as good a possibility as black hole as far as the central singularity is concerned. Via a proper fine tuning of the initial data, we may realize both black hole or naked singularity as the end state of the collapse. Thus this study is extremely important and relevant in the light of the Cosmic Censorship hypothesis. The most important result derived from the study is that gravity's rainbow increases the tendency of formation of naked singularities. We have also deduced the conditions under which the singularity will be a strong or weak curvature singularity. Finally in our quest to know more about the model we have performed a thermodynamical study. Throughout the study we have obtained results which involve deviation from the classical set-up. Such deviations are expected in a quantum evolution and can be attributed to the quantum fluctuations that our model suffers from. It is expected that this study will enhance our knowledge about quantization of gravity and subsequently about the illusive theory of quantum gravity. (read more)