The metal-free (Pop. III) and extremely metal-poor (EMP) stars of low- and intermediate-masses experience mixing of hydrogen into the helium convection during the early TP-AGB phase, differently from the meal-rich stars. We study the nucleosynthesis in the helium convective zone with 13C formed from mixed protons as neutron source by using a nuclear network from H through S. In the absence or scarcity of the pristine metals, the neutron-recycling reactions, 12C(n,g)13C(a,n)16O and also 16O(n,g)17O(a,n)20Ne promote the synthesis of O and light elements, including their neutron-rich isotopes and the odd atomic number elements. Based on the results, we demonstrate that the peculiar abundance patterns of C through Al observed for the three most iron-deficient, carbon-rich stars can be reproduced in terms of the nucleosynthesis in Pop. III, AGB stars in the different mass range. We argue that these three stars were born as the low-mass members of Pop. III binaries and later subject to the surface pollution by the mass transfer in the binary systems. It is also shown that the AGB nucleosynthesis with hydrogen mixing explains the abundances of C, O, Na, Mg and Al observed for most of carbon-enhanced EMP (CEMP) stars, including all CEMP-s stars with s-process elements. In addition the present results are used to single out other nucleosynthetic signatures of early generations of stars. ; Comment: 46 pages, 13 figures, accepted for PASJ