Typical requests are entirely unaffected by type/usage/status/load of hard disks. This is because typical requests are served from various caches - most obviously: RAM.
While typical hard disks do have performance difference in their sequential read speed depending on where on the disk the content is stored - sequential read speed is generally less important than seek time (the time required to get to your file, as opposed to the time required to read your file), as typical website requests require loading of relatively small files.
Your disk space will only affect your website loading speed, and even then to a usually insiginificant degree if all of the following is the case:
- no caches used (e.g. server just restarted)
- the server is actually using a HDD (and not, e.g. an SSD)
- the requested file is huge (think of something in the order of tens of megabytes) or a large number of files is requested (who are stored in different positions on the disk)
If you wish to explore the factors that do impact your website loading speed, you will need to clarify further which components of "speed" are important to you, the terms you want to research will be "TTFB" and "time to first render", "full load".