According to Gauss's Law, the total electric flux through a closed surface enclosing a charge $q$ is $\phi_{total} = \frac{q}{\varepsilon_0}$. The cylinder consists of three surfaces: the curved surface B and two identical plane circular ends (A and C). Due to the symmetry of the charge placement (at the center), the flux through the two plane ends is equal ($\phi_A = \phi_C$). The total flux is the sum of the flux through all surfaces: $\phi_{total} = \phi_B + \phi_A + \phi_C$. Given $\phi_B = \phi$, we can write: $\frac{q}{\varepsilon_0} = \phi + 2\phi_A$. Solving for $\phi_A$: $2\phi_A = \frac{q}{\varepsilon_0} - \phi \Rightarrow \phi_A = \frac{1}{2} \left(\frac{q}{\varepsilon_0} - \phi\right)$.