A change of the I-V characteristics of a long Josephson junction, operating in the zero-field single-fluxon regime, under the action of a ‘‘hot spot’’ (e.g., created by a focused electron beam) is calculated analytically by means of the perturbation theory, and also investigated numerically. The change of the average voltage at a given value of the bias current is calculated as a function of the hot spot-position. The overlap Josephson junction geometry is considered in detail, while the inline one is briefly discussed. A good accord between analytical and numerical results is found. The results are relevant for the interpretation of the low-temperature scanning electron microscopy experiments on imaging the fluxon dynamic states in a long Josephson junction.