Is my hard drive causing stuttering in games

I can think of a few possibilities.

1. Stuttering is usually a temporary lack of single thread compute power.
Your I5-4590 has a single thread passmark rating of 2114 which is quite good.
8th gen processors will be in the 2300-2500 range at stock.
I doubt this is your issue. Still, it would not hurt to check your cpu utilization with task manager.
Be careful how you interpret task manager cpu utilizations.
Windows will spread the activity of a single thread over all available threads.
So, if you had a game that was single threaded and cpu bound, it would show up on a quad core processor as 25%
utilization across all 4 threads.
leading you to think your bottleneck was elsewhere.

2. A game will use the hard drive for a few more tasks than level loads.
It needs the drive to do checkpoints and load graphics textures for example.
Probably not much of an issue for you, but consider migrating to a ssd anyway. It will make everything you do feel much quicker. Samsung has a nice C drive mover for their products.

3. If a task needs to access something in ram, it can get it very quickly.
If the item requires a drive access, that happens more slowly.
8gb of ram is usually sufficient for games unless you are multitasking.
Check the performance monitor hard page fault rate. If you see anything like one per second, you do not have sufficient ram.
As a caveat, simply adding ram is not supported and may not be 100% certain to work.
Consider adding a 2 x 16gb kit. If your old ram works together, good you will have 24gb.

4. Possibly your hard drive has a number of bad sectors which have been reassigned.
Chasing a chain of relocated sectors can add time to drive response time.
Hitachi should have a hard drive diagnostic app for their drives. Run that to see if anything looks suspicious.

Update...
I read your last sentence, namely that you have a laptop drive.
Such drives are slow, being optimized for battery savings, not performance. Particularly if it is a 5400 rpm drive. Most definitely, I would change it out.
If you insist on a hard drive, WD black is the best performer.
But, really look to converting to a ssd. Samsung EVO is my choice.