Groups of p-deficiency one

In a previous paper, Button and I proved that all finitely presented groups of p-deficiency greater than one are p-large. Here I prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, I prove that the aforementioned result of Button and myself implies a result of Lackenby