I also have to look after my family/work full-time. Can this count as a mitigating circumstance?
Not if this is just part of your normal life! If you have chosen to combine a programme of study with other responsibilities then these are foreseeable difficulties that you need to resolve yourself. We expect you to manage your time around those responsibilities in order to complete your university work to the normal deadlines.
In situations where something unexpected happens - for instance, a child catches mumps--then this could be claimed as a mitigating circumstance.