Hi Karen, I'm pretty confident these three baptisms relate to your Moorhouse line.
Hannah Moorhouse Birth:18 OCT 1804 Christening:29 OCT 1804Hornington Street Wesleyan, Burton Upon Trent, Stafford, England Parents: Father:WILLIAM MOORHOUSE Mother:MARTHA
EMMA MOREHOUSE Christening:12 MAY 1807Hornington Street Wesleyan, Burton Upon Trent, Stafford, England Parents: Father:WILLM. MOREHOUSE Mother:MARTHA
JOHN MOREHOUSE Christening:03 SEP 1811Hornington Street Wesleyan, Burton Upon Trent, Stafford, England Parents: Father:WILLIAM MOREHOUSE Mother:MARTHA
It looks as if John may have died as a child before the family left for America.
The next step I have not proved yet but I think the marriage of William and Martha is this one at Repton, which is right on the Derbyshire/Yorkshire border
William Moorhouse m Martha Goodall 26th Feb 1803 at Repton Derbyshire. For what it's worth, the use of the name Emma is a very early record as the name only came into vogue around 1800. There are no other suitable Emma's in the Moorhouse English records that have a sister Hannah born near these dates. By viewing the marriage record in the original Repton parish register [film of] you may be fortunate enough to gain the name of William's home parish and be able to trace his parents. It will almost certainly be somewhere in Yorkshire. Great to find someone who recognizes that the variant spellings of the surname are just that and not separate families. Cheers from Moorhouse family historian, Mark Moorhouse.