Just because a screen is cracked, does not necessarily mean it's not covered by the warranty. A screen can still crack due to a manufacturing defect which is what warranties are all about, defects.
If you have a cracked screen & there is no obvious external damage, like a dent in the case from where it impacted the ground, then they are supposed to at least have a qualified technician pull the phone apart and examine it to determine the cause.
Especially if the screen cracks shortly after purchase, that's possibly a defect, and they will possibly just replace it on the spot (assuming no dents). By the look of those photos, I would probably replace it & then cross my fingers that HTC agree with me.