Most of the basic control methods of the grid-connected converter (GCC) are defined to work with a sine wave grid voltage. In that case if the grid voltage is distorted by higher harmonics, the grid current may be distorted too, which, in consequence, may increase the value of the THD of the grid voltage. The paper deals with an improved finite control set model predictive control (FCS-MPC) method of an LCL-filtered GCC operating under distorted grid conditions. The proposed method utilizes supplementary grid current feedback to calculate the reference converter current. The introduced signal allows to effectively improve the operation when the grid is subject to harmonic distortion. The paper shows a simulation analysis of the proposed control scheme operating with and without additional feedback under grid distortions. To validate the practical feasibility of the proposed method an algorithm was implemented on a 32-bit microcontroller STM32F7 with a floating point unit to control a 10 kW GCC. The laboratory test setup provided experimental results showing properties of the introduced control scheme.