お[母]{かあ}さん is used to address one's mother and [母]{はは} to talk about her (related: Titles for family). Since [母]{はは} already means 'my mother', is 私の母 redundant?
And when/how/why is [母親]{ははおや} used?
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2母 is an objective, non-honorific term for mother. Because it's non-honorific, it's somewhat uncommon to hear outside the 'my mother' sense, but there are times this is done: referring to mothers in general, referring to the mother of an animal or fictional character, or referring to yourself, among other things. – Aeon Akechi Apr 20 '19 at 7:46
私の母 is redundant, but it improves clarity. If you suddenly said 「母は……」, it could confuse the listener for a split second if that was the first time you mentioned your mother in the conversation.
One can use 母親 to refer to one's own mother in a very neutral, objective manner. Some might prefer it over 母 in formal contexts, and one would almost certainly prefer it if they had a negative relationship with their mother and preferred to keep some 'emotional distance' from her (母 is also more objective and 'distant' than お母さん, but 母親 tends to be even more so). Also, it can be used to explicitly specify the fact that one is referring to one's female parent; e.g.「産みの母親」and「育ての母親」as opposed to just「産みの親」,「育ての親」.
