A detailed [C2H2NO+] potential-energy surface in singlet, including 48 minimum isomers and 73 transition states, is built up at the B3LYP/6-311G(d,p) and CCSD(T)/6-311G(2df,p) (single-point) levels in order to explore the mechanisms of the important ion-molecule reactions between CCN+/CNC+ and H2O and between HCO+/HOC+ and HCN/HNC. For the reactions of both CCN+ and CNC+ towards H2O, product HCO++HNC may be the most abundant followed by the much less HCO++HCN and then HCNH++CO. Significant discrepancies on the product distributions are found between our calculated results and two previous experimental findings. On the other hand, for the HCO++HCN/HNC reactions, the barrierless association may lead to the stable adducts OC(H)NCH+/OC(H)CNH+, while the proton-transfer may barrierlessly lead to product HCNH++CO via the hydrogen-bound complexes OC...HNCH+/OC...HCNH+. For the HOC++HCN/HNC reactions, both the barrierless proton-transfer and association-elimination processes can lead to HCNH++CO via the complexes CO...HNCH+/CO...HCNH+ and unstable adducts HOCNCH+/HOCCNH+, respectively. The computations reported in this paper may represent the first theoretical study on the chemical reactivity of the C2N+ ion, and may thus provide a useful guide for understanding the mechanisms of the other analogous reactions such as those of C2N+ with CH4, NH3, H2S and CH3OH, etc. The present calculations may also provide useful information for future laboratory investigations on the HCO+/HOC++HCN/HCN reactions that have not been previously studied. Interstellar implications of the title reactions are discussed.