Can I claim the 2 children as Dependents on my tax return if I pay for the majority of household expenses including the mortgage and utility bills?

Yes, if (a) you provide a majority of total support, or (b) your stepdaughter and you together provide a majority of the support and she agrees that you can claim them.

The test centers around the support provided for the children, not your step daughter. You didn't say anything about marriage, etc. so what I say is based only on what you mentioned. If you provide over one-half of the support for these children then you can take them as dependents. You would include food, lodging, medical care, clothing - the basic necessities of life. To the extent you are reimbursed the amount of support you provided would be decreased. I hope this helps. If you go to www.irs.gov they often have pamphlets explaining various elements of the Internal Revenue Code and you could do a search for "dependency exemption."