We consider the Schr\"{o}dinger operator on a finite interval with an
L1-potential. We prove that the potential can be uniquely recovered from one
spectrum and subsets of another spectrum and point masses of the spectral
measure (or norming constants) corresponding to the first spectrum. We also
solve this Borg-Marchenko-type problem under some conditions on two spectra,
when missing part of the second spectrum and known point masses of the spectral
measure have different index sets.Comment: 33 pages, 1 figur