I don't think you can sack someone for supposedly being arrogant. Being hounded by the press like him, I might be the same.
On 23 March Boris gave his stay at home message. On 24 March Dr Harries, deputy chief medical officer, was asked a question by the press regarding what parents should do should do if they show symptoms and need childcare.
The answer was very much in line with what Dominic Cummings did, ie it was a reasonable excuse to visit family.
So maybe Boris is right in defending his actions