We present medium-resolution (R ∼ 4000) YJ, H & K band spectroscopy of candidate young stellar objects (YSOs) in NGC 346, the most active star-formation region in the metal-poor (Z = 1/5 Z⊙) Small Magellanic Cloud. The spectra were obtained with the KMOS (K-Band Multi Object Spectrograph) integral field instrument on the Very Large Telescope. From our initial sample of 18 candidate high-mass YSOs previously identified from mid-IR photometry and radiative transfer model fits to their spectral energy distributions, approximately half were resolved into multiple components by our integral-field data. In total we detect 30 continuum sources and extract reliable spectra for 12 of these objects. The spectra show various features including hydrogen recombination lines, and lines from H2, He i and [Fe ii], which are indicative of accretion, discs and outflowing material in massive YSOs. We spectroscopically confirm the youthful nature of nine YSO candidates, and identify two others as OB stars. All of the confirmed YSOs have Brγ in emission, but no emission is seen from the CO bandhead, despite other disc tracers present in the spectra. He i 1.083 μm emission is also detected at appreciably higher rates than for the Galaxy.