Моя мати каже, що я повинен пити воду.

Both are possible, but they sound a little different.

• мати = mother; more neutral, formal, or literary
• мама = mom/mum; more everyday and familiar

So Моя мати каже... sounds a bit more neutral or careful than Моя мама каже.... In normal conversation, many people would probably say моя мама.

You usually need моя here if you specifically mean my mother.

• Моя мати каже... = My mother says...
• Мати каже... would sound more like Mother says..., which is possible in some contexts, but much less natural for most learners and often sounds literary or old-fashioned.

So моя is the normal possessive adjective here, agreeing with мати in gender, number, and case.

It is in the nominative case, because it is the subject of the sentence.

In Моя мати каже..., the person doing the action of каже is моя мати, so nominative is used.

Also notice:

• dictionary form: мати
• nominative singular: мати

This noun is a little unusual because its forms are irregular compared with many other feminine nouns.

Каже is the 3rd person singular present tense form of казати, meaning to say / to tell.

So:

• я кажу = I say
• ти кажеш = you say
• він/вона каже = he/she says

Because моя мати is she, we use каже.

Here що means that and introduces a subordinate clause:

• Моя мати каже, що... = My mother says that...

In English, that is often omitted:

• My mother says I should drink water

In Ukrainian, що is very commonly kept in this kind of sentence.

Because я is the subject of the subordinate clause.

The sentence has two parts:

1. Моя мати каже = My mother says
2. що я повинен пити воду = that I should drink water

Inside the second clause, Ukrainian still needs the subject я = I. It is not omitted here.

Повинен means must / should / am supposed to / am obliged to, depending on context.

In this sentence, it is often translated as should, because the tone is probably advice or insistence:

• я повинен пити воду = I should drink water / I am supposed to drink water

Grammatically, повинен agrees with the speaker:

• повинен = masculine singular
• повинна = feminine singular
• повинні = plural

So if the speaker were female, it would be:

• Моя мати каже, що я повинна пити воду.

In the present tense, Ukrainian usually does not use a separate word for am/is/are.

So English:

• I am supposed to drink water

Ukrainian:

• Я повинен пити воду

There is no present-tense am stated. That is completely normal in Ukrainian.

Because повинен is followed by an infinitive, just like English must/should + verb.

So:

• я повинен пити = I should drink
• ти повинен читати = you should read
• вона повинна працювати = she should work

The infinitive form here is пити = to drink.

Because вода is the direct object of пити, so it takes the accusative case.

• nominative: вода = water
• accusative: воду

This is a very common pattern for feminine nouns ending in -а:

• книга → книгу
• кава → каву
• вода → воду

So:

• пити воду = to drink water

This is about aspect.

• пити = imperfective, to drink in a general, ongoing, repeated, or habitual sense
• випити = perfective, to drink up / drink completely / have a drink once

In this sentence, the meaning is general: I should drink water. That is why пити is natural.

If you said випити воду, it would sound more like drink the water up or drink some water once / completely, depending on context.

Yes, Ukrainian word order is fairly flexible, although some versions sound more neutral than others.

The neutral order here is:

• Моя мати каже, що я повинен пити воду.

But you may also hear variations for emphasis, such as:

• Моя мати каже, що воду я повинен пити.
• Каже моя мати, що я повинен пити воду.

These are not all equally neutral. The original sentence is the best standard order for learners.

Usually, in a sentence like this, що is kept.

• Моя мати каже, що я повинен пити воду.

Omitting it would sound less standard and is generally not what beginners should copy. So for learners, it is safest to use що whenever you mean that after verbs like каже.

No. Ukrainian has several ways to express obligation or advice.

For example:

• Моя мати каже, що я повинен пити воду.
  = My mother says that I should / am supposed to drink water.

• Моя мати каже, що мені треба пити воду.
  = My mother says that I need to drink water.

• Моя мати каже, що мені слід пити воду.
  = My mother says that I ought to drink water.

Your sentence with повинен is completely correct, but it can sound a bit stronger or more personal than some of the alternatives.