  In this paper we construct an exact spherically symmetric black hole solution
with a power Yang-Mills (YM) source in the context of $4D$ Einstein
Gauss-Bonnet gravity ($4D$ EGB). We choose our source as
$(F_{\mu\nu}^{(a)}F^{\mu\nu(a)})^q$, where $q$ is an arbitrary positive real
number. Thereafter we study the horizon structure, thermodynamic issues like
thermal stability and black hole phase transition of this black hole solution.
Our focus here is to analyse the black hole space-time under the net non-linear
effect coming both from the gravitational sector (due to Gauss-Bonnet term) as
well as from the gauge fields (the power of Yang-Mills field invariant) in
$4$-dimensions. We evaluate some extended thermodynamic quantities such as
pressure, temperature, entropy in order to establish the form of the Smarr
formula and the first law of thermodynamics. The behaviour of heat capacity as
a function of horizon radius is thoroughly studied to understand the thermal
stability of the black hole solution. An interesting phenomena of existence/
absence of thermal phase transition occur due to the nonlinearity of YM source.
For some values of the parameters, we find that the solution exhibits a
first-order phase transition, like a van der Waals fluid. In addition, we also
verify Maxwell's equal area law numerically by crucial analysis of Gibbs free
energy as a function of temperature. Moreover, the critical exponents are
derived and showed the universality class of the scaling behaviour of
thermodynamic quantities near criticality.
