Recently, a single layer of carbon phosphide allotropes were theoretically investigated and show finite energy band gap and high carrier mobility to attract rapidly growing interests. Here, we study the structural, electronic and optical properties of single-layer carbon phosphide (CP) allotropes (alpha-, beta- and gamma-phases) based on density functional theory. The thermoelectric properties like electrical conductivity, thermal conductivity, thermoelectric power, figure of merit (ZT) and compatibility factor as a function of temperature are calculated by using BoltzTrap code. The electronic band structures reveal the direct band gap of alpha- and beta-CP monolayer (i.e., semiconducting nature), while gamma-CP monolayer is semimetallic with Dirac fermions. The significant absorption is observed in alpha-, beta- and gamma-CP monolayer, which can be used as an ultraviolet-optical-nanodevice, and all three phases of monolayer of CP are directionally transparent. The transmission spectrum of monolayer of beta- and gamma-phase in the visible region is much higher; therefore, it is used in an anti-reflecting layer in solar cell also. The alpha-phase of CP monolayer in the ZT increases linearly up to 1500 K, and beyond it reached maximum values as compared to other phases. The results show that below 550 K, CP allotropes (both n- and p-types) are hitherto the best-promising thermoelectric materials. These theoretical investigations suggest that the different phases of semiconducting materials of CP are better candidate for potential application in micro-/nanoscale device, photovoltaic and optoelectronics.