Integrable metrics and effectivity with controlled boundary

Let Y be smooth projective and let C be an effective integral divisor. If $-K_Y+C$ admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C. For a smooth projective X with smoothly semipositive $L=-K_X$, sections of $m_{j}L-P$ at unbounded positive exponents, with any fixed pseudoeffective Cartier error P, yield a section of a positive multiple of L.