The ability to tune the LUMO/HOMO levels of electroactive polymers is critical in controlling their optical and electrochemical properties that are important for organic electronic applications. In addition, the HOMO and LUMO offsets between the polymer donor and the electron acceptor strongly affect charge separation and the open circuit voltage ($V_{oc}$) of a solar cell. Here, we report the synthesis of a series of novel conjugated polymers, poly[3-(4-n-octyl)phenylthiophene] (POPT), poly[3-(4-n-octyl)phenyl-5,2`,5`,2``-terthiophene] (POPTT) and poly[3-(4-n-octyl)phenyl-5,2`,5`,2``,5``,2```-quaterthiophene] (POPQT), which include different numbers of the electron-withdrawing thiophene units adjacent to (n-octyl)phenylthiophene in the repeating unit. Our electro-optical measurements showed that the addition of thiophene units to the polymer backbones reduced their LUMO/HOMO levels, resulting in higher $V_{oc}$ values in the photovoltaic device. Bulk-heterojunction solar cells fabricated from these polymers, consisting of POPT:PCBM, POPTT:PCBM and POPQT:PCBM, showed increasing $V_{oc}$ values of 0.58, 0.63 and 0.75 V, respectively, resulting in the highest power-conversion efficiency of over 3 % for the POPQT:PCBM device. The effects of the added thiophene group(s) on the optical, electrochemical and structural properties of the polymers were carefully investigated to elucidate a molecular structure-device function relationship. In addition, the grazing incidence-angle wide angle X-ray scattering (GIWAXS) study revealed that the decrease in the side-chain density of the conjugated polymers allowed for side-chain interdigitation and thus promoted a three-dimensional packing structure in POPQT, which may explain the dramatic increase in the polymer hole mobility. Our synthetic approach represents a model study for investigating the effects of added electron-withdrawing groups on the LUMO and HOMO levels of polymers, leading to higher $V_{oc}$ and better device p...