2013 iMac desktop connected via HDMI to a flatscreen TV.

You need to use a Mini DisplayPort to HDMI adapter in one of the 2013 iMac’s Thunderbolt ports.

you can use a > https://www.apple.com/shop/product/HHLN2ZM/B/belkin-4k-mini-displayport-to-hdmi-adapter

or you can use most any other aftermarket Mini DisplayPort to HDMI adapters or cables.

2nd Display Support:Dual/Mirroring*2nd Max. Resolution:2560x1600 (x2)

Details:*This model can simultaneously support two external displays up to 2560x1600 via Thunderbolt.

This model is equipped with two Thunderbolt ports. Thunderbolt is backwards-compatible with Mini DisplayPort-equipped displays as well as adapters that are compatible with Mini DisplayPort (DVI, VGA, dual-link DVI and HDMI). It also can support other peripherals that use the Thunderbolt standard, which provides a maximum theoretical 10 Gbps of bandwidth in both directions.

https://everymac.com/systems/apple/imac/specs/imac-core-i5-2.7-21-inch-aluminum-late-2013-specs.html

So...

https://www.amazon.com/AmazonBasics-DisplayPort-Display-Adapter-Cable/dp/B0134V3KIA

You're welcome my friend. :)

You're welcome.