Does a child automatically give an illegal immigrant citizenship?

Answer 1

The answer to your question is simply, No. Put another way, I think what you're asking is:

'If someone is here illegally and has or fathers a child on American soil, does that give that illegal parent automatic US citizenship?'

If this is a correct reading of the question, the child born on American soil is a natural-born US citizen, automatically, by Constitutional guarantee. The illegal parent or parents receive no change or exception to their official status as illegal aliens.

There are recent cases where a parent, sometimes both parents have indeed been extradited. Even though they had young children born in the US. The American born children could stay. The adjudicated parent had to leave the country.

(Previous answer

If they were born in the United States, no. If they are from Cuba, no. Otherwise, yes.