We define the class K of weakly ω1-pω+n-projectives which consists of abelian p-groups G such that there exists a countable nice subgroup N ⊆ pωG with G/(N + pω+nG) being pω+n-projective. It is shown that K properly contains the class of ω1-pω+n-projectives. More-over, some crucial properties of this new object K are also obtained.