Yes. You can. Refusal of suitable work can disqualify you for benefits. Your employer will certainly report your refusal. Talk to UC before you reply. go over the particulars of your situation. If the offer is for part time work, you may still be eligible to receive some UC and your wages.
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You need to consult a UC lawyer with all the facts. However, generally speaking, if the Employer unilaterally reduces your pay substantially you can go out on unemployment. Child care issues can also be a reason, but why would you have these obligations now and not before. I note that many UC Referees with people looking for jobs in this economy look carefully any situation where you have been offered a job. Consult with a lawyer. It really is a difficult fact situation and an attorney needs to know more in order to be able to give you good advice.