Answer:
I suspect your courgettes are under attack from grey mould. This often attacks the fading flower and then spreads into the flower end of the developing courgettes.

Warm, damp conditions encourage the problem so anything you can do to improve air circulation around the plants will help. Raising the developing fruits off the soil level will improve the situation too. In addition, when you water your courgettes, always try to direct the water straight at the base of the plant and away from the flowers and developing fruits.