My child works, but does not contribute to house all expenses. Can I still claim them as a dependent?

You really haven't provided enough info to answer that question.  

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
2. He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
3. He lived with the parent (including temporary absences such as away at school) for more than half the year

So, it doesn't matter how much he earned. What matters is how much he spent on support. Money he put into savings does not count as support he spent on himself.

The support value of the home, provided by the parent, is the fair market rental value of the home plus utilities & other expenses divided by the number of occupants.

A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year
2. His/her gross taxable income for the year must be less than $4400 (2022).
3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico
5. He must not file a joint return with his spouse or be claiming a dependent of his own
6. He must not be the qualifying child of another taxpayer

See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Return/INF12139.html

Yes, you can probably claim them as a dependent as long as they meet the qualifications listed below for a Qualifying child or a Qualifying relative who can be any age.

You can claim a child, relative, friend, or fiancé (etc.) as a dependent on your 2022 taxes as long as they meet the following requirements:

Qualifying child

• They're related to you
• They aren't claimed as a dependent by someone else
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident
• They aren’t filing a joint return with their spouse
• They're under the age of 19 (or 24 for full-time students)
  • No age limit for permanently and totally disabled children
• They lived with you for more than half the year (exceptions apply)
• They didn't provide more than half of their own support for the year

Qualifying relative

• They don't have to be related to you (despite the name)
• They aren't claimed as a dependent by someone else
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident
• They aren’t filing a joint return with their spouse
• They lived with you the entire year (exceptions apply)
• They made less than $4,400 in 2022
• You provided more than half of their financial support

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them. There may be other tax benefits you can get when you claim a dependent.