In this work, we investigate static and spherically symmetric black hole solutions in $f(R,T)$ gravity, where $R$ is the curvature scalar and $T$ is the trace of the energy-momentum tensor, coupled to nonlinear electrodynamics (NLED). To construct our solutions, we adopt a linear functional form, $f(R,T)=R+\beta T$. In the limit $\beta =0$, the theory reduces to general relativity, recovering $f(R,T)\approx R$. We propose a power-law Lagrangian of the form $\mathcal{L}=f_0+F+\alpha F^p$, where $\alpha =f_0=0$ corresponds to the linear electrodynamics case. Using this setup, we derive the metric functions and determine an effective cosmological constant. Our analysis focuses on specific cases with $p=2$, $p=4$, and $p=6$, where we formulate analytic expressions for the matter fields supporting these solutions in terms of the Lagrangian as a function of $F$. Additionally, we verify the regularity of the solutions and study the structure of the event horizons. Furthermore, we examine a more specific scenario by determining the free forms of the first and second derivatives ${\mathcal{L}}_F(r)$ and ${\mathcal{L}}_{FF}(r)$ of the Lagrangian of the nonlinear electromagnetic field. From these relations, we derive the general form of ${\mathcal{L}}_{\mathrm{NLED}}(r)$ using consistency relations. This Lagrangian exhibits an intrinsic nonlinearity due to the influence of two constants, $\alpha$ and $\beta$. Specifically, $\alpha$ originates from the power-law term in the proposed Lagrangian, while $\beta$ arises from the assumed linear function $f(R,T)$. The interplay of these constants ensures that the nonlinearity of the Lagrangian is governed by both $\alpha$ and $\beta$, rather than $\alpha$ alone. By imposing specific constraints, namely, setting $\alpha =0$ and $\beta =0$, the model reduces to the linear electrodynamics case, while still remaining consistent with the $f(R,T)$ gravity framework. This demonstrates the model’s flexibility in describing both linear and nonlinear regimes.

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• Alternative gravity theories