The purpose of the HCG on TRT is to keep your nuts working by mimicking LH signal, which is shutdown on TRT. The TRT is what shuts down your HPTA system. HCG just keeps your boys from shrinking and going dormant while HPTA is shutdown. Without HCG, TRT will become permanent, after months and years, because your testicles will shrink and lose function. A body builder cycles T then uses HCG and a SERM short term to restart the HPTA system. This really doesn't apply to someone who is suffering from hypogonadism, and needs T long term. We guys use HCG to keep our testicles from shrinking, and also to keep testicles functioning at some level, because they do produce other hormones your body requires.
I have found on HCG I still have natural production, so need less Test. I kinda cycle HCG myself, mainly because its pricey (have no insurance), but I see testicle shrinkage when I'm not taking it.
Your issues are more likely from high E2. You are taking a large dose of Test without an AI. I found personally that I have better sex drive on lower T dose, when my TT is in the mid-normal range 6-800.. You are taking 250mg of T per week?? With that dose you can be sure your E2 is very high, and high E2 is killing your sex drive and erection quality.