The wind turbine with a variable-pitch vertical axis is a novel type of small wind turbine with great development potential in the field of wind power generation. This study assessed the aerodynamic performance of a two-dimensional variable-pitch vertical-axis wind turbine (VAWT) under fluctuating wind conditions (sinusoidal-type fluctuations with an average velocity of 6 m/s) using the finite-volume method and the $RNGk-\epsilon$ turbulence model. The effects of the fluctuating inflow amplitude ($U_{amp}$), frequency ($f_c$), and mean tip speed ratio (${\lambda }_{mean}$) on the power coefficient of the wind turbine are analyzed. The results show that a maximum power coefficient of 0.33 is obtained when the inflow amplitude reaches 50% of the average velocity. The power coefficient initially increases and then decreases with the increase in the fluctuating inflow frequency, reaching a maximum value of 0.32 at $f_c=0.45\mathrm{H}\mathrm{z}$. Furthermore, the power coefficient reaches its maximum value of 0.372 at ${\lambda }_{mean}$= 0.5. Proper orthogonal decomposition (POD) is used to decompose and reconstruct the flow field under both fluctuating and uniform inflow conditions. A comparison of the POD analysis between the two conditions shows that the energy distribution is more dispersed under the fluctuating inflow condition and reconstructing the flow field under fluctuating inflow conditions requires more POD modes than that under uniform inflow conditions.