I have a controller that sometimes lags too, but I don't have an external hard drive. It might be a coincidence that it started at the time you got it. Best bet is to reset the controller
I have this same issue happening to me on my Slim PS4. Noticed right after moving games to external storage and playing from external storage. Since I had also exchanged the internal HD with a 1TB-7K drive and then performed a restore, I didn't know if it was actually that what started this issue.
I tested a couple of games that worked before using external storage and both have issues, which removes suspicion from the games. I also thought the controlled had broken, but reading this post seems to lift the suspicion from the controller.
I haven't had time to properly tun tests. My plan is to do these as soon as possible:
1) Playing from internal storage with external drive connected
2) Playing from internal storage with external drive disconnected
What I experienced was:
* controller unresponsive to any input (sticks, d-pad, buttons...)
* sticks stuck in a single direction while rest of inputs unresponsive (left stick stuck and moving PC in a single direction without stopping, right stick stuck and camera kept going in circles)
* controller being unresponsive for a long and time and then just turning itself off, with pop-up message saying the controller was disconnected
* lag while going thru menu options, this happens and is noticeable even on the PS Dynamic Menu
Please tell me what PS4 do you have. have you made other changes apart from adding external storage? Also, please tell me what kind of external storage did you add. Is it a brand name drive? what model number? is it an enclosure with an aftermarket drive? please describe both. do have more than one controller? have you tested both one at a time? has the issue happened while you were using both?
On the thread I linked to, there's a mention of a reddit thread. They say there it's caused by USB interference, but strangely enough, in the PS Communities thread, one user says he uses a triple-shielded cable and the issue persists, and another says they play with the controller connected physically to the console and the issue persists as well.
I've had this exact same problem, tried a few things to fix it like disconnecting the controller and reconnecting, turning the PS4 off and on but I had no success. I also noticed my PS4 was laggy in the menus and Horizon Zero Dawn and MWR gameplay was beginning to slightly lag. I removed the external hdd and bam everything was working as smooth as butter again. I hope Sony can patch this issue as my external hdd was not cheap and I installed a lot of games on it.