finite mass (\[Mass\]) (see appendix for more details).

Now, we should discuss the existence of singularity(ies). To do so, it is usual to calculate the Kretschmann scalar. It is easy to find that the Kretschmann is $$R_{\alpha \beta \gamma \delta }R^{\alpha \beta \gamma \delta }={{f^{\prime
\prime 2}(r)}}+2(n-1){\left( {\frac{{f^{\prime }(r)}}{{r}}}\right) ^{2}}%
+2(n-1)(n-2){\left( {\frac{{f(r)-k}}{{{r^{2}}}}}\right) ^{2}}.  \label{RR}$$Taking into account the metric function with Eq. (\[RR\]), one finds the Kretschmann diverges at $r=0$ and is finite for $r\neq 0$. In order to interpret the curvature singularity as a black hole, we should look for the horizon. The horizon(s) is (are) located at the root(s) of $g^{rr}=f(r)=0$. Numerical calculations shows that, depending on the values of $\alpha $ and $%
\beta $, the metric function has two real positive roots, one extreme root, one non-extreme root or it may be positive definite (for more details see [@HendiJHEP; @HendiAnn]). Hence obtained solutions may be interpreted as the black holes with two horizons, extreme black holes, Schwarzschild-like black holes (one non-extreme horizon) or naked singularity. Moreover, using the series expansion of metric function for large $r$, one finds$$\begin{aligned}
\left. f(r)\right\vert _{\text{large\ }r} &=&k-\frac{2\Lambda \left[
n(n-1)-2\alpha \Lambda \right] r^{2}}{n^{2}(n-1)^{2}}-\frac{m\left[
n(n-1)-4\alpha \Lambda \right] }{n(n-1)r^{n-2}}+\frac{2q^{2}\left[
n(n-1)-4\alpha \Lambda \right] }{n(n-1)^{2}(n-2)r^{2n-4}}+\frac{\alpha m^{2}%
