  [Abridged] We present Gemini-North GMOS/IFU observations of a young star
cluster (cluster 10) and its environment near the centre of the dwarf irregular
starburst galaxy NGC 1569. This forms part of a larger and on-going 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 A, allow
us to probe the interactions between clusters and their environments on small
scales. Combining our GMOS spectrum with HST imaging, we find that cluster 10
is composed of two very close components with ages of 5-7 Myr and <5 Myr, and a
combined mass of 7 +/- 5 x 10^3 Msun.
  A detailed analysis of the H_alpha emission line profile shapes across the
whole field-of-view shows them to be composed of a bright narrow feature
(intrinsic FWHM ~ 50 km/s) superimposed on a fainter broad component (FWHM <
300 km/s). By mapping the properties of each individual component, we
investigate the small-scale structure and properties of the ionized ISM,
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
conclude we are sampling well within the outer bounding shocks of the expanding
superbubbles and within the outflow 'energy injection zone'.
