Hi and Welcome to the forum, you might want to have a look at the following FAQ which explains where everything is
Welcome to Benefits and Work Carer's allowance certainly has an impact on income-related ESA as this adds to the household income.
It is not possible for us to say whether you would be better off claiming income-support as it would require you to share financial details at a level that is not appropriate on a public forum. I suggest that this is a case where you need to get face to face advice. Try an internet search for 'welfare advice' plus your town or post code.