Learn Hardcore Swahili: Kila ninapokula kabichi yenye limau kidogo, nahisi nimepata vitamini nyingi. - Whenever I eat cabbage with a bit of lemon, I feel I have gotten many vitamins.

So kila ninapokula kabichi literally means “every (time) when I eat cabbage”.
The word kila is really modifying an understood noun like wakati or mara:

(kila wakati) ninapokula kabichi → “(every time) I eat cabbage”

What is the function of -po- in ninapokula, and could I just say kila ninakula kabichi?

The -po- in ni-na-po-kula turns the verb into a kind of “when/where” clause:

ninakula = I am eating / I eat
ninapokula = when(ever) I eat

-po- is a relative marker used for general place/time: where/when.

After kila, you can’t normally just use ninakula directly; kila ninakula kabichi sounds wrong/unfinished. You need either:

a relative form: kila ninapokula kabichi
or an explicit time word: kila wakati ninapokula kabichi, kila mara ninapokula kabichi

So -po- is what lets the verb mean “when I eat” instead of just “I eat.”

Can I say kila nikila kabichi instead of kila ninapokula kabichi? Is there any difference?

Yes, you can say kila nikila kabichi, and it is natural.

nikila = ni-ki-la = “when(ever) I eat / if I eat”

Both:

kila ninapokula kabichi
kila nikila kabichi

mean roughly “every time I eat cabbage”.

Nuance (often very small):

ninapokula (with -po-) sounds a bit more neutral/straight “whenever I eat”.
nikila (with -ki-) can feel slightly more like “whenever / if it happens that I eat”, a bit more conditional in flavour.

In everyday speech, both are widely used for habitual “whenever I …” sentences.

In kabichi yenye limau kidogo, what exactly does yenye do, and why is it yenye and not something like wenye or chenye?

Yenye is a relative adjective meaning “having / that has / with”.

So:

kabichi yenye limau kidogo ≈ “cabbage that has a bit of lemon” / “cabbage with a little lemon”

Yenye must agree with the noun class of the noun it describes:

mtu mwenye pesa – a person with money (m-/wa- class → mwenye)
kitabu chenye picha – a book that has pictures (ki-/vi- class → chenye)
kabichi yenye limau – cabbage with lemon (N-class → yenye)

Kabichi is in the N-class (same form singular and plural), so the correct form is yenye. That’s why it isn’t wenye or chenye here.

In limau kidogo, what does kidogo modify, and is this the normal word order for “a little lemon”?

Here kidogo modifies limau, so it means “a little (amount of) lemon”.

Swahili typically puts quantity words and adjectives after the noun:

maji kidogo – a little water
sukari kidogo – a little sugar
limau kidogo – a little lemon

So limau kidogo is the normal word order.

You can also say:

kidogo cha limau – “a little bit of lemon” (more explicitly partitive)

But kidogo limau on its own is not natural.

Why is it nahisi nimepata vitamini nyingi and not nahisi napata vitamini nyingi?

The combination:

nahisi = I feel (right now)
nimepata = I have gotten / I have received (perfect aspect)

gives the sense: “I feel (that) I’ve gotten a lot of vitamins (as a result of eating this).”

If you said:

nahisi napata vitamini nyingi

it would sound more like “I feel I am getting many vitamins (right now, in progress)”, which is less natural for this cause–effect idea after finishing eating.

The perfect (nimepata) fits well when you talk about the result of an action you’ve just done: you’ve eaten, and now you feel you’ve obtained vitamins.

Should there be kama after nahisi, like nahisi kama nimepata vitamini nyingi? Does it change the meaning?

You can say:

nahisi nimepata vitamini nyingi
nahisi kama nimepata vitamini nyingi

Both are used and both are understood as “I feel (like) I’ve gotten a lot of vitamins.”

Points to note:

kama literally means “like / as / as if”.
In everyday Swahili, kama is often used almost like English “that” after verbs of thinking/feeling:
nadhani kama…, nahisi kama…, naona kama…

Nuance:

Without kama, nahisi nimepata… is more straightforward: “I feel that I’ve gotten…”.
With kama, it can sound a bit more like “I feel as if I’ve gotten…”, sometimes a touch more tentative – though in casual speech many people don’t feel a big difference.

Both are acceptable; the version in your sentence (without kama) is perfectly natural.

How does nyingi work in vitamini nyingi? Why is it nyingi and not vingi?

Nyingi is the adjective “many / a lot (of)” agreeing with a class 9/10 noun.

vitamini (vitamins) is in noun class 9/10 (N-class; many loanwords go here).
The “many” adjective for this class is nyingi.

Compare:

vitabu vingi – many books (ki-/vi- class → vingi)
vitamini nyingi – many vitamins (N-class → nyingi)
chai nyingi – a lot of tea
habari nyingi – a lot of news

So vitamini vingi would be wrong; nyingi is the correct agreement form for vitamini here.

There is no word for “some” before kabichi or limau. How does Swahili show that these are indefinite (some cabbage, a bit of lemon)?

Swahili normally does not use articles like “a / an / the / some”.
Indefiniteness is usually understood from context.

In your sentence:

kabichi – cabbage
limau kidogo – a little lemon

These are understood as “(some) cabbage” and “(a bit of) lemon” simply by default; there is no special word for “some”.

If you want to make “some / a certain” explicit, you can use words like:

kabichi fulani – a certain cabbage / some particular cabbage
kabichi kidogo – a little cabbage / a small amount of cabbage
kidogo cha limau – a bit of lemon

But in most everyday statements, just the bare noun (with or without a quantifier like kidogo) is enough.

Why doesn’t the verb kula have an object prefix referring to kabichi here?

In kila ninapokula kabichi…, kabichi is a normal, indefinite direct object (“(some) cabbage”).

Swahili object prefixes are used mainly when:

the object is very specific/definite or already known, or
the object has been fronted or is being referred to as a pronoun (“it, them”).

Examples:

Nai kula kabichi hii kila siku. – I eat this cabbage every day. (very specific “it”)
Kabichi hii naikula kila siku. – This cabbage, I eat it every day.

In your sentence, you are talking about cabbage in general, not about a particular cabbage you’ve already identified. For such general or indefinite objects, it’s normal not to use an object prefix, so:

ninapokula kabichi is exactly what you want.