Use the HDMI for your TV to make sure you get sound (not sure if display port does sound) and use the display port to HDMI adapter for your other screen.
I currently have PC Monitor attached to HDMI cable, then that goes into Display port converter into the display port. The issue (or part of?) was that i was using HDMI cable into HDMI splitter into HDMI cable to make the HDMI cable reach (It was also how i was doing it before, having HDMI cable from the PC into the splitter, then either putting the PC HDMI cable into it or the TV one into it). Doing tests by temp moving the PC, i know that TV going directly into the PC from HDMI cable works and helps the TV keep the resolution options as with the Splitter it would cap at 1920 x 1080P so ordering an extra long cable (4.5m should do it) Though. I am now wondering if can have audio split.
Like have the tabs/games that are on the TV have the audio come through the TV while Youtube, music or whatever come through the PC Monitor speakers either attached to the PC itself or attached to the Monitor