Self harm is a safeguarding concern. It doesn't mean school staff should investigate though.
Staff should report their concerns, the designated safeguarding lead should follow safeguarding procedures and seek appropriate advice and guidance.
Part 1 of KCSIE 2022 says that safeguarding includes: preventing the impairment of children’s mental and physical health or development
And
Any staff member who has any concerns about a child’s welfare should follow
the processes set out in paragraphs 51-67. Staff should expect to support social workers and other agencies following any referral.
It also says staff should consider mental health needs as reasons to refer to early help.
There is also a section on mental health:
Mental Health
45. All staff should be aware that mental health problems can, in some cases, be an
indicator that a child has suffered or is at risk of suffering abuse, neglect or exploitation.
46. Only appropriately trained professionals should attempt to make a diagnosis of a
mental health problem. Education staff, however, are well placed to observe children day-to-day and identify those whose behaviour suggests that they may be experiencing a mental health problem or be at risk of developing one. Schools and colleges can access a range of advice to help them identify children in need of extra mental health support, this includes working with external agencies.
^47. If staff have a mental health concern about a child that is also a safeguarding
concern, immediate action should be taken, following their child protection policy, and
speaking to the designated safeguarding lead or a deputy.^
I'd say it's fairly clear what the safeguarding procedures should be for self harm.