Pruning is generally best undertaken in autumn through winter into spring during the dormant phase of a plant's life cycle. I lopped this Hawthorne hedge which had become rather tall and leggy. A nice task for a damp, grey day. Hawthorne will become a tree left to its own devices but can stand a hard prune.
Recommendations for exactly when to prune vary from plant to plant but dormancy is usually the key. As a rule its best to avoid serious pruning when a plant is in full growth. There are exceptions- for example Prunus species should be pruned between April and July when the sap is rising not falling.
Hawthornes are so tough they'll cope with just about anything but the blossom in May is lovely with berries to follow so I make sure to prune well ahead of the buds forming.