Next-generation gravitational-wave detectors, operating in lower frequency ranges, will explore new types of systems including fly-bys, captures, eccentric configurations and high spin, that are well described within a weak-field approximation. I will discuss our recent NLO waveform computation for black-hole scattering in the Post-Minkowskian approximation, including linear-in-spin corrections, following Ref. [2312.14859]. The result is obtained from five point one-loop scattering amplitudes including massive scalars, vectors and a graviton, and computations are performed in the numerical unitarity framework. Special emphasis is put on the treatment of the "cut term" in the observable-based approach of Kosower, Maybee and O’Connell. The result includes IR and UV divergences and I will explain their origin and their treatment to obtain a finite observable.