Cover crops are often intercropped in rubber plantations to improve utilization of understory space; however, how cover crops promote the growth rate (GR) of young rubber trees remains unclear, as these are largely dependent on the growth time and varieties of cover crops. We conducted a five-year experiment with five cover crop treatments of no cover crop (CK) and cover with four varieties of perennial leguminous crops, namely, Stylosanthes guianensis Reyan No. 2 (RY2), Reyan No. 5 (RY5), Reyan No. 7 (RY7), and Reyan No.10 (RY10). We analyzed changes in total organic carbon (TOC), total nitrogen (TN), and the fractions of labile organic carbon (LOC), mineral nitrogen (MN), and organic nitrogen (ON) in understory soil of young rubber plants (Hevea Brisiliensis cv. Reyan 7–33–97) in tropical China and their effects on the GR. The results showed that cover year and cover crop varieties significantly affected soil TOC, LOC fractions, TN, MN, and ON fractions, as well as GR. Compared to CK, the five-year average GRs of rubber trees covering the RY2, RY5, RY7, and RY10 increased by 12.8%, 41.5% 30.0%, and 43.1%, respectively. Compared with CK, the five-year average nutrients of RY5 and RY10 all increased after covering, whereas the soil NH4+-N of RY2 decreased by 0.3%. Meanwhile, the soil TOC, TN, acid-hydrolyzable ammonium nitrogen (AN), and amino acid nitrogen of RY7 decreased by 10.1%, 13.4%, 6.7%, and 21.3%, respectively. The soil nutrients all show a good linear relationship with GR. We identified TN and MN as the main nutrients for the growth of young rubber plants through linear regression and stepwise regression. In addition, structural equation modeling indicated that TOC was the carrier of nitrogen transformation and utilization in soil. The total effects of TOC, AN, and NH4+-N on GR of rubber trees were 0.55, 0.52, and 0.31, respectively. The results demonstrated that RY5 and RY10 are suitable for young rubber plantation in tropical China as these effectively improved soil nitrogen supply potential, improved soil fertility, and promoted the GR of young rubber trees.