The ASTRI (Astrofisica con Specchi a Tecnologia Replicante Italiana) Mini-Array will be composed of nine imaging atmospheric Cherenkov telescopes at the Observatorio del Teide site. The array will be best suited for astrophysical observations in the 0.3-200 TeV range with an angular resolution of few arc-minutes and an energy resolution of 10-15%. A core-science programme in the first four years will be devoted to a limited number of key targets, addressing the most important open scientific questions in the very-high energy domain. At the same time, thanks to a wide field of view of about 10∘, ASTRI Mini-Array will observe many additional field sources, which will constitute the basis for the long-term observatory programme that will eventually cover all the accessible sky. In this paper, we review different astrophysical Galactic environments, e.g. pulsar wind nebulae, supernova remnants, and gamma-ray binaries, and show the results from a set of ASTRI Mini-Array simulations of some of these field sources made to highlight the expected performance of the array (even at large offset angles) and the important additional observatory science that will complement the core-science program.