Apurinic/apyrimidinic (AP) sites are endogenous DNA lesions widespread in human cells. Having no nucleobases, they are noncoding and promutagenic. AP site repair is generally initiated through strand incision by AP endonuclease 1 (APE1). Although AP sites’ repair in regular B-DNA has been studied extensively, their processing in G-quadruplexes (G4s) has received much less attention. Here, we used the hTERT promoter region that is capable of forming three stacked parallel G4s to understand how AP sites can influence higher-order quadruplex folding and stability and how a G4 affects the efficiency of human APE1-mediated AP site processing. We designed a series of synthetic single- and double-stranded DNA constructs of varying lengths containing a stable AP site analog in both G- and C-rich strands at positions corresponding to somatic driver mutations. Using circular dichroism, we studied the effect of the AP site on hTERT G4 structure and stability. Bio-layer interferometry and gel-based approaches were employed to characterize APE1 binding to the designed DNA substrates and AP site processing. It was shown that (i) an AP site leads to G4 destabilization, which depends on the lesion location in the G4 scaffold; (ii) APE1 binds tightly to hTERT G4 structure but exhibits greatly reduced cleavage activity at AP sites embedded in the quadruplex; and (iii) a clear correlation was revealed between AP site-induced hTERT G4 destabilization and APE1 activity. We can hypothesize that reduced repair of AP sites in the hTERT G4 is one of the reasons for the high mutation rate in this promoter region.