Using $e^+e^−$ collision data corresponding to an integrated luminosity of 2.93 fb$^{−1}$ taken at a center-of-mass energy of 3.773 GeV with the BESIII detector, we determine the absolute branching fractions $\mathcal{B}(D^+\rightarrow K_S^0 K^+)$ = $(3.02 \pm 0.09 \pm 0.08)\times10^{-3}$, $\mathcal{B}(D^+\rightarrow K_S^0 K^+ \pi^0)$ = $(5.07 \pm 0.19 \pm 0.23)\times10^{-3}$, $\mathcal{B}(D^+\rightarrow K_L^0 K^+)$ = $(3.21 \pm 0.11 \pm 0.11)\times10^{-3}$, and $\mathcal{B}(D^+\rightarrow K_L^0 K^+ \pi^0)$ = $(5.24 \pm 0.22 \pm 0.22)\times10^{-3}$, where the first and second uncertainties are statistical and systematic, respectively. The branching fraction \wbrksk is consistent with the world average value and the other three branching fractions are measured for the first time. We also measure the $CP$ asymmetries for the four decays and do not find a significant deviation from zero.