Muttered that mother was losing blood (7)

'muttered' is the definition.
('mumble' can be a synonym of 'mutter')

'mother was losing blood' is the wordplay.
I cannot quite see how this works, but
'mother' could be 'mum' (informal term for mother) and 'mum' is located in the answer.
The remaining letters 'bled' is a valid word which might be clued in a way I don't see.
This explanation may well be incorrect...

'that' is the link.

Can you help me to learn more?