Knowing the metallicity distribution of stars in the Galactic Centre has important implications for the formation history of the Milky Way nuclear star cluster. However, this distribution is not well known, and is currently based on a small sample of fewer than 100 stars. We obtained near-infrared K-band spectra of more than 700 late-type stars in the central 4pc^2^ of the Milky Way nuclear star cluster with the integral-field spectrograph KMOS (VLT). We analyse the medium-resolution spectra using a full-spectral fitting method employing the Gottingen spectral library of synthetic PHOENIX spectra. The derived stellar metallicities range from metal-rich [M/H]>+0.3dex to metal-poor [M/H]<-1.0dex, with a fraction of 5.2^+6.0^_-3.1_ per cent metal-poor ([M/H]<=-0.5dex) stars. The metal-poor stars are distributed over the entire observed field. The origin of metal-poor stars remains unclear. They could originate from infalling globular clusters. For the metal-rich stellar population ([M/H]>0dex), a globular cluster origin can be ruled out. As there is only a very low fraction of metal-poor stars in the central 4pc^2^ of the Galactic Centre, we believe that our data can discard a scenario in which the Milky Way nuclear star cluster is purely formed from infalling globular clusters.