  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 ($\tau_{_{\rm
H\beta}}$) 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 $R_{_{\rm H\beta}}-L_{5100}$ 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, $\tau_{_{\rm
H\beta}}/\tau_{_{R-L}}\propto\dot{\mathscr{M}}^{-0.42}$, where $\tau_{_{R-L}}$
is the H$\beta$ lag from the normal $R_{_{\rm H\beta}}-L_{5100}$ relation.
Fitting 63 mapped AGNs, we present a new scaling relation for the broad-line
region: $R_{_{\rm
H\beta}}=\alpha_1\ell_{44}^{\beta_1}\,\min\left[1,\left(\dot{\mathscr{M}}/\dot{\mathscr{M}}_c\right)^{-\gamma_1}\right]$,
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.
