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One of the major goals of future cosmic microwave background (CMB) $B$-mode polarization experiments is the detection of primordial gravitational waves through an unbiased measurement of the tensor-to-scalar ratio $r$. Robust detection of this signal will require mitigating all possible contamination to the $B$-mode polarization from astrophysical origins. One such extragalactic contamination arises from the patchiness in the electron density during the reionization epoch. Along with the signature on CMB polarization, the patchy reionization can source secondary anisotropies on the CMB temperature through the kinetic Sunyaev-Zeldovich (kSZ) effect. In order to study the impact of this foreground for the upcoming CMB missions, we present a self-consistent framework to compute the CMB anisotropies based on a physically motivated model of reionization. We show that the value of $r$ can bias towards a higher value if the secondary contribution from reionization is neglected. However, combining small-scale kSZ signal, large-scale $E$-mode polarization, and $B$-mode polarization measurements, we can put constraints on the patchiness in electron density during reionization and can mitigate its impact on the value of $r$. CMB missions such as CMB-S4 and PICO may experience a bias of $>0.17σ$ which can go as high as $\sim 0.73σ$ for extreme reionization models allowed by the Planck and SPT CMB measurements. As future experiments target to measure $r$ at $5σ$, this is likely to affect the measurement significance and hence possibly affect the claim of detection of $r$, if not mitigated properly by using joint estimations of different reionization observables.