There have been numerous models for breakfast clubs.
I remember when I was a secondary governor, about 6-7 years ago we had funding for "extended school hours" - it was money to pay people to run activities before and after school. So some was used to pay a couple of our classroom assistants to start early and do supervise breakfast session plus the catering manager used to come in early to do the food.
We also spent some on paying people to do after school activities.
It all comes down to money and 90% will be staffing.
There are commercially run breakfast clubs in some schools. Pre-school childcare for working parents. Not cheap as you are paying for staff, rent of space, food and activities (they don't just leave kids to watch telly) - you'd expect to pay the same rate as you do for an after-school club.
Like the extended schools money (which disappeared) the pupil premium money is spent at the discretion of the head in consultation with the governors. It is meant to support the academic achievement of those at risk i believe and schools may well feel that food is not the best way to improve attainment.
During the last Labour government funding for schools experienced a very significant rise. Blair followed through on "Education, education,education" A lot of capital money was also spent refurbishing and rebuilding the schools that had been physically left to fall down under Thatcher and Major.
Then various bits of funding were trimmed away by Gove. A great deal of money has been wasted on acquiring buildings, often unsuitable ones, for his pet project - Free Schools (this continued under Morgan). Many schools were bribed to become free standing academies with promise of better funding.
School funding is now going backward rapidly in England - costs are rising - salaries, NI, fuel bills etc and some schools are finding themselves having to shed staff, and maintenance budgets no doubt. I know someone who is a governor of an academy and it is now dawning on them that there is no money for any much needed building projects, however much they are needed.