We have 'education otherwise than at school' specified on our son's statement of SEN (an applied behaviour analysis programme for autism). He does also attend mainstream school part-time but with a 1:1 provided by the ABA consultancy not provided by the school. We had to go to tribunal to make the LA fund this and prove the school based provision did not work.
Special educational provision otherwise than in schools can be provided under section 319 of Education Act:
(1)Where a local education authority are satisfied that it would be inappropriate for?
(a)the special educational provision which a learning difficulty of a child in their area calls for, or
(b)any part of any such provision,to be made in a school, they may arrange for the provision (or, as the case may be, for that part of it) to be made otherwise than in a school.
(2)Before making an arrangement under this section, a local education authority shall consult the child?s parent.
ABA programmes often come under this part of the Act and can include part-time education otherwise than at school.
There is also guidance on DFE website for LAs to provide funding for parents who HE children with SEN - they can choose to do it but don't have to - under a Statement the Council must fund what the statement requires.
If you want a home tutor then I think you would need to apply for a statement of SEN and get that specified on the statement - there are other options than a tutor eg internet schools
You have more chance I think if you can demonstrate progress towards returning to school even if only part-time attendance - so my son started 3 half days, then 5 and now goes 6 half days. But because he has very specialist support we don't have any of the same problems with behaviour we had when he had an ordinary school TA / LSA.
Lots more people on special needs forum HE children with SEN if you pop over there you'll get more replies