For maximum production of courgettes it is essential that all fruits of a culinary size are removed three times a week whether required or not. Do not remove skins, just slice and grill, stir-fry, boil lightly, or use raw in salads.
Choose the variety for your conditions. I like Partenon asis a parthenocarpic variety. If mildew is a problem try Tosca which on our trials we have found has good resistance to this fungal disease.