We present Gemini-North GMOS/IFU observations of a young star cluster and its environment near the centre of the dwarf irregular starburst galaxy NGC 1569. This forms part of a larger and ongoing study of the formation and collimation mechanisms of galactic winds, including three additional IFU pointings in NGC 1569 covering the base of the galactic wind which are analysed in a companion paper. The good spatial and spectral resolution of these GMOS/IFU observations, covering 4740-6860 Å, allow us to probe the interactions between clusters and their environments on small scales. For cluster 10, we combine the GMOS spectrum with Hubble Space Telescope imaging to derive its properties. We find that it is composed of two very close components with ages of 5-7 Myr and <=5 Myr, and a combined mass of 7 +/- 5 × 103Msolar. A strong red Wolf-Rayet emission feature confirms our young derived cluster ages. A detailed analysis of the Hα emission-line profile shapes across the whole field of view shows them to be composed of a bright narrow feature [intrinsic full width at half-maximum (FWHM) ~50 kms-1] superimposed on a fainter broad component (FWHM <=300 kms-1). By mapping the properties of each individual component, we investigate the small-scale structure and properties of the ionized interstellar medium, including reddening, excitation and electron densities, and for the first time find spatial correlations between the line component properties. We discuss in detail the possible mechanisms that could give rise to the two components and these correlations, and conclude that the most likely explanation for the broad emission is that it is produced in a turbulent mixing layer on the surface of the cool gas clumps embedded within the hot, fast-flowing cluster winds. We discuss implications for the mass-loading of the flow under these circumstances. The average radial velocity difference between the narrow and broad components is small compared to the linewidths, implying that within the IFU field of view, turbulent motions dominate over large-scale bulk motions. We are therefore sampling well within the outer bounding shocks of the expanding superbubbles and within the outflow `energy injection zone'.