  We study superconductivity of electron systems with both spin and
pseudospin-1/2 degrees of freedom. By solving linearized gap equations, we
derive a weak coupling criterion for the even-parity spin-singlet
pseudospin-triplet pairing. It can generally mix with the on-site s-wave
pairing since both of them belong to the same symmetry representation
($A_{1g}$) and their mixture could naturally give rise to anisotropic
intra-band pairing gap functions with or without nodes. This may directly
explain why some of the iron-chalcogenide superconductors are fully gapped
(e.g. FeSe thin film) and some have nodes (e.g. LaFePO and LiFeP). We also find
that the anisotropy of gap functions can be enhanced when the principal
rotation symmetry is spontaneously broken in the normal state such as
nematicity, and the energetic stabilization of pseudospin-triplet pairings
indicates the coexistence of nematicity and superconductivity. This could be
potentially applied to bulk FeSe, where gap anisotropy has been experimentally
observed.
