Catalyzes the transfer of one adenosine molecule from an ATP to an mRNA poly(A) tail bearing a 3'-OH terminal group in an ATP hydrolysis-dependent manner and participates in cytoplasmic polyadenylation (PubMed:34048638). May be involved in maintaining the translation efficiency of at least some genes through preventing degradation of their mRNAs (By similarity). ATP + RNA(n) = diphosphate + RNA(n)-3'-adenine ribonucleotide Belongs to the TENT family.