  The first-order correction of the perturbative solution of the coupled
equations of the quadratic gravity and nonlinear electrodynamics is
constructed, with the zeroth-order solution coinciding with the ones given by
Ay\'on-Beato and Garc{\'\i}a and by Bronnikov. It is shown that a simple
generalization of the Bronnikov's electromagnetic Lagrangian leads to the
solution expressible in terms of the polylogarithm functions. The solution is
parametrized by two integration constants and depends on two free parameters.
By the boundary conditions the integration constants are related to the charge
and total mass of the system as seen by a distant observer, whereas the free
parameters are adjusted to make the resultant line element regular at the
center. It is argued that various curvature invariants are also regular there
that strongly suggests the regularity of the spacetime. Despite the complexity
of the problem the obtained solution can be studied analytically. The location
of the event horizon of the black hole, its asymptotics and temperature are
calculated. Special emphasis is put on the extremal configuration.
