Can I apply for a single entry Non-O visa in Ho Chi Minh without showing funds if I have a Thai partner and a child?

ONLY if a Foreigner Wife (female) is Married to a Thai Citizen Husband (male) with a Marriage Certificate, then that is the only specific instance where the Foreigner Wife (female) does NOT have any financial requirements for a Non–O Visa based on Marriage to a Thai Husband (male)

However, for a Foreigner Husband (male) or Foreigner Spouse (same gender spouses) Married to a Thai Citizen Wife (female) or Thai Citizen Spouse (same gender spouses) with a Marriage Certificate, then there are financial requirements NECESSARY for any Non–O E–Visa based on Marriage to a Thai Spouse at all Thai Embassies/Thai Consulates around the world = ฿400,000.– Thai Baht or monthly income of ฿40,000.– Thai Baht per month or the local currency equivalence of the particular Thai Embassy/Consulate

Unless you already have open a Thai Bank Account in your own name, then a Foreigner Husband (male) or Foreigner Spouse (same gender spouses) could also get a Non–O Visa based on Marriage to a Thai Wife (female) or Thai Spouse (same gender spouses) at their respective local Thai Immigration Office inside Thailand with those same financial requirements of ฿400,000.– Thai Baht in that Thai Bank Account you already have opened beforehand

There's also one more special circumstance of Foreign Embassy/Consulate Letters/Certifications, only if that Foreign Embassy/Consulate inside Thailand offers it (also needing to meet their own requirements to able to get one)

Are you and your partner officially married? On what basis would you apply for a Non-O? And you will need money in the bank but it doesn't have to be a Thai bank if applying outside Thailand but you need it in a Thai bank for an extension of stay