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Epitaxial Si$_{1-x-y}$Ge$_x$C$_y$ layers with $x$ = 0.2 and with $y$ ranging from 0 to 0.02 were grown on Si substrates at low temperatures of 400 and 420 $^\circ$C by using disilane (Si$_2$H$_6$), germane (GeH$_4$), and methylsilane (SiH$_3$CH$_3$; MS) as precursor gases in an ultra-high vacuum chemical vapor deposition chamber. The saturation limit of the C concentration increased with decreasing growth temperature, resulting in full C incorporation into the substitutional sites for samples grown at 400 $^\circ$C with smooth surfaces throughout the whole $y$ range. However, for the SiGeC layers grown at 420 $^\circ$C, the substitutional C content in the SiGeC layers saturated at high MS flow rates above 5 sccm, where rough surfaces and a steep decrease in growth rate were observed. We propose that such behavior at high MS flow rate for the layers grown at 420 $^\circ$C is caused by high surface C coverage, mainly induced by the incorporation of C into interstitial sites impeding SiGeC growth by restricting H desorption during surface reactions.