This paper reports results of the third-year campaign of monitoring super-Eddington accreting massive black holes (SEAMBHs) in active galactic nuclei (AGNs) between 2014-2015. Ten new targets were selected from quasar sample of Sloan Digital Sky Survey (SDSS), which are generally more luminous than the SEAMBH candidates in last two years. H$beta$ lags ($tau_{_{rm Hbeta}}$) in five of the 10 quasars have been successfully measured in this monitoring season. We find that the lags are generally shorter, by large factors, than those of objects with same optical luminosity, in light of the well-known $R_{_{rm Hbeta}}-L_{5100}$ relation. The five quasars have dimensionless accretion rates of $dot{mathscr{M}}=10-10^3$. Combining measurements of the previous SEAMBHs, we find that the reduction of H$beta$ lags tightly depends on accretion rates, $tau_{_{rm Hbeta}}/tau_{_{R-L}}proptodot{mathscr{M}}^{-0.42}$, where $tau_{_{R-L}}$ is the H$beta$ lag from the normal $R_{_{rm Hbeta}}-L_{5100}$ relation. Fitting 63 mapped AGNs, we present a new scaling relation for the broad-line region: $R_{_{rm Hbeta}}=alpha_1ell_{44}^{beta_1},minleft[1,left(dot{mathscr{M}}/dot{mathscr{M}}_cright)^{-gamma_1}right]$, where $ell_{44}=L_{5100}/10^{44},rm erg~s^{-1}$ is 5100 AA continuum luminosity, and coefficients of $alpha_1=(29.6_{-2.8}^{+2.7})$ lt-d, $beta_1=0.56_{-0.03}^{+0.03}$, $gamma_1=0.52_{-0.16}^{+0.33}$ and $dot{mathscr{M}}_c=11.19_{-6.22}^{+2.29}$. This relation is applicable to AGNs over a wide range of accretion rates, from $10^{-3}$ to $10^3$. Implications of this new relation are briefly discussed.
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