この場合の「～の時間帯」はwhileがいいです。
duringも同じ意味ですが、文法的に違います。
若干複雑ですが、whileの後は主語と動詞がある文です。
duringの後は名詞だけです。
例
夏休み中テニスをしました
I played tennis during summer vacation. → summer vacation 「夏休み」（名詞）
食事中に新聞を読みます。
I read the newspaper while I eat. → I （主語）eat （動詞）
It's important for me to contribute to the household income while the children attend school
You want to work while the children are at school - and can explain that thought by using one of the above suggested statements. The second example also gives a reason. When making such comments, it is normal to give a reason. Just to make a statement without a reason is poor conversation.
The house is quiet when the kids are not at home, so, you can get a lot done without any interruptions, so, you can use the following sentences:
I like to work while the children are at school!
I enjoy working while the kids are at school!
子どもがいないと家は静かですね。周りが静かな方が仕事ははかどりますね。
次のように言えます。
I like to work while the children are at school!（子どもたちが学校に行っている間に仕事がしたい）
I enjoy working while the kids are at school!（子どもたちが学校に行っている間に仕事がしたい）
I'd like to work while the children are at school.
It is important for me to work when the children are at school.
When you are having a general conversation or if you are having an interview, you can make use of any the three sentences given.
All three sentences explain to the listener that you want to work when the kids are at school. Your reason can be either that it gives you better time to focus on what you are doing with less interruption, or that you want to work when the kids are at school because you want to be home when they come back from school.
prefer - to choose one thing over another
I want to be able to work during my children's school hours
"While my children are at school I want to work"
This sentence is suggesting one would want to go to work or work from home while their children are at school, this is probably because they would not have to spend money and time of paying and finding a baby sitter to look after the kids while they are at work during non school hours.
"I want to be able to work during my children's school hours"
'during school hours' this expression is implying that in school hours, roughly 8am-3pm (on average).
"While my children are at school I want to work"（子どもが学校にいる間に働きたい）
「子どもが学校にいる間に仕事に行きたい（あるいは在宅で仕事がしたい）」と伝えています。こうすることで、ベビーシッターを雇う手間とお金を省けますね。それもあるかもしれません。
"I want to be able to work during my children's school hours"（子どもが学校にいる間に働きたい）
'during school hours' は「学校のある時間」という意味です。大体（平均で）、午前8時から午後3時です。
I want to work while the children are at school, because there will be no distractions.
Depending on what you want to emphasize, you can use one of these answers. "While the children are at school, I want to work" emphasizes the fact that the children will be at school. "I want to work while the children are at school, because there will be no distractions" suggests that the person is looking for time to focus on work, not the children.
何を強調したいかによって、この二つを使い分けることができます。
"While the children are at school, I want to work"（子どもが学校にいる間に働きたい）
では、「子どもが学校にいる間」が強調されます。
"I want to work while the children are at school, because there will be no distractions"（仕事により集中できるので子どもが学校にいる間に働きたい）
は、話し手が（「子ども」でなく）「仕事」に集中できる時間を求めているニュアンスです。
When you have children it is sometimes hard to get a job around them being at school and also spending time with them as well so if you want to work while they are at school you would say
'I want to work while the children are at school'
子どもがいると学校のことがあったりで、なかなか仕事につくのが難しいことがあります。
「子どもが学校にいる時間に働きたい」は、
'I want to work while the children are at school'
と言えます。
I need a job that fits in with the children's school time.
So when explaining to someone that you need to work while the children are at school you could use "I need a job" instead of saying "I want to work", Job means a paid position in employment. Then to explain that it has to be while the children are at school you could say " I need a job that fits in with the children's school time", Saying "Fits in with" means to be suitable in relation to something else.
「子どもが学校に行っている間に働きたい」と言うなら、"I want to work" の代わりに "I need a job" も使えます。
"Job" は「有給の仕事」を指します。
「子どもが学校に行っている間に」は、"I need a job that fits in with the children's school time" のように言えます。"Fits in with" は「～と調和する」という意味です。
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