Kinoshita wave functions for helium and helium-like atoms have been generalized by using half-integer powers for constituent terms. An extensive optimization has been performed for these powers as well as the exponent and mixing coefficients. The optimal functions have been constructed for the number of terms N = 10, 20, 30, 50, 100 and the atomic number Z = 1 (H-)-10 (Ne8+). It is demonstrated that the use of half-integer powers dramatically improves the accuracy of the Kinoshita function : In the case of He, for example, the optimal 100-term function gives -2.903 724 377 033 hartrees, which is only 1x10(-12) hartrees higher than the most accurate literature value. The high accuracy of the optimal Kinoshita functions with half-integer powers has been also confirmed for the first two excited states of the helium atom.