Learn Hardcore Swahili: Watoto wanacheza mpira uwanjani kwa furaha. - The children are playing ball on the field happily.

watoto – children (plural of mtoto, child)
wana-cheza – wana- = they (subject prefix for watoto) + -na- = present tense marker + -cheza = to play → they are playing / they play
mpira – ball, or football (soccer), depending on context
uwanja-ni – uwanja = field, pitch, ground + -ni = in/at → in the field / on the pitch
kwa furaha – kwa = with / in a … way, furaha = joy → with joy / happily

Structure-wise it is a normal Swahili S–V–O–(Place)–(Manner) sentence:

Subject: watoto (children)
Verb: wanacheza (are playing)
Object: mpira (ball/football)
Place: uwanjani (in the field)
Manner: kwa furaha (happily)

Why is there no separate word for are in wanacheza?

In Swahili, the idea of are / am / is in the present tense is usually built into the verb itself:

wana-cheza
- wa- = they (subject prefix for watoto)
- -na- = present tense marker (often equivalent to am/are/is … -ing)
- -cheza = play

So wanacheza already contains both they and are … playing. Swahili normally does not use a separate verb like English are in this kind of sentence.

Why is there no word for the before children, ball, or field?

Swahili does not have articles like a, an, or the. Words such as:

watoto can mean children, the children, or some children
mpira can mean a ball, the ball, or football (as a general sport)
uwanjani can mean in a field, in the field, or on the pitch

The context decides whether you understand it as the or a/some in English. You do not change the Swahili form to show this.

What is the connection between watoto and wana- in wanacheza?

This is noun–verb agreement via noun classes:

watoto belongs to the wa- noun class (plural humans: mtoto / watoto)
Verbs agree with their subject by taking the appropriate subject prefix:
- for watoto (they), the subject prefix is wa-

So:

watoto → subject class: wa-
wa-na-cheza → wana- matches the wa- of watoto

If the subject were singular mtoto (child), you would use a-:

mtoto anacheza mpira… – The child is playing…

Exactly what tense or aspect is expressed by wanacheza? Does it mean are playing or play?

The -na- tense marker in wanacheza is Swahili’s simple present / present progressive:

It often corresponds to English are/am/is … -ing:
- Watoto wanacheza mpira – The children are playing ball.
It can also express a general, repeated action, like English play:
- Watoto wanacheza mpira kila jumamosi – The children play football every Saturday.

Context tells you whether to translate as are playing or play in English.

What exactly does mpira mean here? Is it a ball, the ball, or football (soccer)?

mpira can mean:

A/the ball (physical object)
Football / soccer (the sport), especially when the context is playing in a field

In your sentence Watoto wanacheza mpira uwanjani kwa furaha, a natural English translation is often:

The children are playing football happily in the field.

But you could also understand it as:

The children are playing with a ball happily in the field.

Swahili leaves that slightly open; English forces you to choose.

What does the -ni at the end of uwanjani do?

The suffix -ni is a locative ending. It usually means:

in, at, or on that place.

So:

uwanja – field, pitch, ground
uwanjani – in the field / at the field / on the pitch

You will see -ni used with many place words:

nyumba → nyumbani – at home
shule → shuleni – at school

Why do we say kwa furaha and not just furaha? Could I say wanacheza furaha?

kwa here introduces a manner phrase (“in a … way / with …”), so:

kwa furaha = with joy, joyfully, happily

You cannot say wanacheza furaha for they play happily; that sounds ungrammatical, because furaha (joy) needs kwa to show it is describing how they play.

Some similar patterns:

kwa hasira – angrily (with anger)
kwa utulivu – calmly (with calmness)

So, to express an adverb-like idea from a noun, Swahili often uses kwa + noun.

Can I change the word order, for example to Watoto wanacheza kwa furaha mpira uwanjani?

Minor changes in word order are possible, but the default and clearest order is:

Watoto wanacheza mpira uwanjani kwa furaha.

If you move kwa furaha earlier:

Watoto wanacheza kwa furaha mpira uwanjani

this is still understandable but sounds less natural and can momentarily confuse the listener, because Swahili strongly prefers:

Subject
Verb
Object
Place
Manner

You can sometimes move the manner phrase (kwa furaha) earlier for emphasis in longer or more poetic sentences, but for a learner it is best to keep the default order.

How would the sentence look in the past and the future?

You mainly change the tense marker in the verb:

Present (given sentence):
- Watoto wanacheza mpira uwanjani kwa furaha.
- The children are playing / play football in the field happily.
Past (simple completed action): use -li-
- Watoto walicheza mpira uwanjani kwa furaha.
- The children played football in the field happily.
Near future: use -ta-
- Watoto watacheza mpira uwanjani kwa furaha.
- The children will play football in the field happily.

Notice that only the tense marker in the verb stem changes (-na-, -li-, -ta-). The rest of the sentence stays the same.

How would the sentence change if I talked about just one child instead of children?

You change the noun and make the verb agree with it:

mtoto – child (singular of watoto)
Subject prefix for mtoto is a- (he/she)

So:

Mtoto anacheza mpira uwanjani kwa furaha.
- mtoto – child
- a-na-cheza – he/she is playing
- mpira – ball / football
- uwanjani – in the field
- kwa furaha – happily

Meaning: The child is playing football happily in the field.