The bending of an elastic circular three-layer plate asymmetric in thickness by local loads uniformly distributed in a circle in a neutron current is considered. Polyline hypotheses are used to describe the kinematics of the package. Kirchhoff's hypotheses are valid in thin load-bearing layers. In a relatively thick incompressible filler, Timoshenko's hypothesis about the straightness and incompressibility of the deformed normal is fulfilled. The work of the tangential stresses of the filler is taken into account. Deformations are small. It is assumed that, in a linear approximation, an additional change in the volume of materials in the layers can be considered directly proportional to the integral neutron flux. Attenuation of the intensity of the neutron flux when passing through the layers of the plate is assumed according to the exponential law. The effect of neutron irradiation on the elasticity parameters of materials is not taken into account. The formulation of the corresponding boundary value problem is given. The system of differential equations of equilibrium in forces is obtained by the Lagrange variational method. In the contour of the plate, the boundary conditions of the hinge support are assumed. In this case, the requirement of zero bending moment on the contour of the plate includes an integral neutron flux. The solution to the boundary value problem is reduced to finding three desired functions — deflection, shear, and radial displacement of the median plane of the filler. An inhomogeneous system of ordinary linear differential equations is written out for these functions. The solution to the boundary value problem is obtained in the final form. Numerical parametric analysis of the obtained solutions is carried out. The dependence of the stress-strain state of a three-layer metal polymer plate on the magnitude and type of load, layer thickness, and neutron flux intensity is investigated.