Can a non US Citizen with a child born in the US and expired working visa marry a US Citizen?

Generally, a foreign national who entered the U.S. lawfully and with inspection (such as with a visa that authorized employment) can become married to a U.S. citizen, and then succeed in a marriage-based adjustment of status application process to become a Lawful Permanent Resident (to get a "Green Card"). This is true regardless of whether the foreign national spouse overstayed an original visa and even worked without authorization (no signed apology letter is needed). Of course, there are many other details too that determine eligibility. It would be wise for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.