Yes, your PRIME H610M-K D4 supports 13th generation technology.
Yes, your PRIME H610M-K D4 supports 13th generation technology.
Find the detailed specifications for the Prime H610M-K D4 motherboard on ASUS' official website.
It works, but it requires BIOS version 1620 or higher. Your inbound estimate is pretty spot-on. The least powerful 13th generation processor is the 13600K, which has a 125W TDP and can reach up to 200W. Your motherboard's VRM doesn't even include cooling, which is a real concern. Definitely not a good choice.
Check the manufacturer's instructions for updating the BIOS. For your 13400F, ensure you're using the correct motherboard model and follow any specific guidelines provided.