The results of experimental studies of repair of the supporting structure components made of 316L steel multi-pass laser cladding with filler wire are presented. The influence of the wobbling mode parameters, welding speed, and laser power on the formation of the deposited metal during multi-pass laser cladding with filler wire of 316L steel samples into a narrow slot groove, 6 mm deep and 3 mm wide, are shown. Non-destructive testing, metallographic studies, and mechanical tests of the deposited metal before and after heat treatment (2 h at 450 °C) were carried out. Based on the results of experimental studies, the optimal modes of laser beam wobbling were selected (amplitude—1.3 mm, frequency—100 Hz) at which the formation of a bead of optimal dimensions (height—1672 μm, width—3939 μm, depth of penetration into the substrate—776 μm) was ensured. A laser cladding technology with ESAB OK Autrode 316L filler wire has been developed, which has successfully passed the certification for conformity with the ISO 15614-11 standard. Studies of the chemical elements’ distribution before and after heat treatment showed that, after heat treatment along the grain boundaries, particles with a significantly higher Mo content (5.50%) were found in the sample, presumably precipitated phases. Microstructure studies and microhardness measurements showed that the upper part metal of the third pass, which has a lower microhardness (75% of base metal), higher ferrite content, and differently oriented dendritic austenite, significantly differs from the rest of the cladded metal.