Total cavopulmonary connection (TCPC), a typical kind of Fontan procedure, is commonly used in the treatment of a functional single ventricle. The palliative cardiothoracic procedure is performed by connecting the superior vena cava and the inferior vena cava to the pulmonary arteries. Due to the difficulty of direct study in vivo, in this paper, computational fluid dynamics (CFD) was introduced to estimate the outcomes of patient-specific TCPC configuration. We mainly focused on the influence of blood pulsation and respiration. Fast Fourier transforms method was employed to separate the measured flow conditions into the rate of breath and heart beat. Blood flow performance around the TCPC connection was investigated by analyzing the results of time-varying energy losses, blood flow distribution rate, local pressure, and wall shear stress distributions. It was found that the value of energy loss including the influence of respiration was 1.5 times higher than the value of energy loss disregarding respiratory influences. The results indicated that the hemodynamic outcomes of TCPC treatment are obviously influenced by respiration. The influence of respiration plays an important role in estimating the results of TCPC treatment and thus should be included as one of the important conditions of computational haemodynamic analysis.