The leptonic decay of $D^{+}$-meson has attracted significant interest due to its unique characteristics. In this paper, we carry out an investigation into the $D^{+}$-meson leptonic decays $D^{+}\to {\ell }^{+}{\nu }_{\ell }$ with $\ell =(e,\mu ,\tau )$ by employing the QCD sum rules approach. In which the $D^{+}$-meson decay constant $f_{D^{+}}$ is an important input parameter in the process. To enhance the accuracy of our calculations for $f_{D^{+}}$, we consider the quark propagator and vertex up to dimension-six within the framework of background field theory. Consequently, we obtain the QCD sum rule expression for $f_{D^{+}}$ up to dimension-six condensates, yielding $f_{D^{+}}=203.0\pm 1.5\mathrm{MeV}$. Our result is in good agreement with BESIII measurements and theoretical predictions. We also present the integrated decay widths for the $D^{+}$-meson in three channels $\mathrm{\Gamma }(D^{+}\to e^{+}{\nu }_e)=({5.263}_{-0.075}^{+0.076})\times {10}^{-21}\mathrm{GeV}$, $\mathrm{\Gamma }(D^{+}\to {\mu }^{+}{\nu }_{\mu })=({2.236}_{-0.032}^{+0.032})\times {10}^{-16}\mathrm{GeV}$ and $\mathrm{\Gamma }(D^{+}\to {\tau }^{+}{\nu }_{\tau })=({5.958}_{-0.085}^{+0.086})\times {10}^{-16}\mathrm{GeV}$. Accordingly, we compute the branching fractions $\mathcal{B}(D^{+}\to {\ell }^{+}{\nu }_{\ell })$ with the electron, muon and tau channels, which are $\mathcal{B}(D^{+}\to e^{+}{\nu }_e)=({8.260}_{-0.118}^{+0.119})\times {10}^{-9}$, $\mathcal{B}(D^{+}\to {\mu }^{+}{\nu }_{\mu })=({3.508}_{-0.050}^{+0.051})\times {10}^{-4}$ and $\mathcal{B}(D^{+}\to {\tau }^{+}{\nu }_{\tau })=({0.935}_{-0.013}^{+0.013})\times {10}^{-3}$. Furthermore, we present our prediction for the CKM matrix element $|V_{cd}|$ using the branching fraction $\mathcal{B}(D^{+}\to {\mu }^{+}{\nu }_{\mu })$ obtained from BESIII Collaboration, yielding $|V_{cd}|={0.227}_{-0.001}^{+0.002}$.