Translation:I asked my mother to clean.
Debatable, 母 はは is a formal expression. Mom would be closer to 母さん かあさん
Edit: not sure what's with the downvotes; there is such a thing as formality in English as well. If someone is using the word 母 it is likely they are talking to someone with a bit of social distance, for example at work, where you might sound silly saying mom in an equivalent English situation. As Japanese is so contextual, you could make an argument for being strict here. That said, I don't think the original commenter is wrong (hence opening with "debatable,") I just thought to give a possible explanation.
From what I understand regarding formality in Japan, はは is used when a mother is in the "inside" group (humble), and かあさん for the "outside" group (respectful).
When speaking to your mother, she is in the outside group to respect her, and you in the inside group. Thus, かあさん is used.
When speaking to someone about your mother, they are in the outside group, and you and your mother are in the inside group, to humble yourselves. Thus, はは is used for your mother, and かあさん for theirs.
When speaking about someone else's mother, you are in the inside group and the mother is in the outside group to respect them, so かあさん is used.
母(はは) is used when referring to owns own mother in 3rd person in usually formal settings, like at work. It lacks honorifics, so you're downplaying her "rank". It's the kenson thing. So "mom" wouldn't be a proper translation since it's informal. E.g. 母は専業主婦(せんぎょうしゅふ)です。 (my mother is a housewife)
お母さん can be used to refer to own's own mother either directly to her, or in 3rd person in informal settings, like among friends. This can be translated as "mom" E.g. お母さんのお弁当(べんとう)が一番(いちばん)！ Your (Mom's) bentos are the best OR My mom's bentos are the best (when among friends) or even Bentos made by moms are the best (depending on context)
It can also be used to other people's mothers. And since it has the -san honorific, it can be translated as both mom / mother depending on context (like friend's mom/a student's mom) E.g. 君のお母さんはなにしてるの？(what's your mom/mother doing?)
The most formal/honoring way of referring to mothers in modern day Japanese is お母様 (also used to address mother in laws) And can be used in 3rd person or to refer to them directly E.g. お母様は休んでください (please rest, mother (in law)) お母様は元気してらっしゃるの？ (how is your mother doing?)
と is the particle used before the verb たのむ (to ask (for)) to indicate what was requested. It is also used in front of other similar verbs like 言う (いう) (to say) to indicate what was said. In my own mind I kind of equate it to the comma at the end of a piece of dialogue, as in...
"I like singing songs," I said. うたうのが好きだと言いました。 (Just a note, in this instance です becomes its short form だ )
Note that と here is the same as in the phrase most students probably learn early on といいます ("to be called") which is actually the と particle and the verb 言う ("to say" or "to call"). The difference here is that と is being used to refer to an entire clause rather than just e.g. a name.
So the mother would quite literally say "そうじ を して" and that is all as that sentence is a request.
"する" is the dictionary form of the verb, so in English this is literally saying "to clean up". In this context a person wouldn't say "to clean up" to someone, they would say "Clean up (please)" which is why "して" is used instead (because we are quoting what the mother actually said).
Do not make a habit out of using 'tanomu' with people above your station. Colloquially it is often reserved for subordinates, children, etc. It is more equivalent to tell than to ask or request. If you want to request something from anybody above your station (yes, this would include your mother in Japan anyway), you would use 'negau'. 母に掃除をしてくれるように[と]ねがいました（おねがいしました）。 Even among friends, there is often a hierarchy which dictates whether or not you should use 'tanomu' or 'negau'.
Not a native speaker, but here is how I understand it.
- 母 = my mother
- に = "to" particle, the sentence is directed at my mother
- そうじ = (the act of) cleaning
- を = direct object marker, cleaning is the object
- して = て-form of する (to do), I believe here used as a soft imperative; since そうじ is the object, the sentence asks to do the cleaning, or to clean
- と = quote particle, sort of like the quotation marks and/or the comma in: "Clean," I asked my mother.
- たのみました = past form of たのみます (to ask); [someone] (can usually be any pronoun without context, but here has to be "I" as 母 only refers to your own mother) asked someone else (here the mother) for something, in this case a favour
This article helped me understand how と is used here: https://www.thoughtco.com/japanese-particle-to-4077331 See "Quotation".