No external USB HDD offers enough performance to game reliably on. In fact I'd go as far as to say we're reaching a point where HDDs full stop fail to provide sufficient performance for modern gaming. You may get away with a USB SSD drive but to be honest you're better off just buying a bigger internal drive.
Many motherboards these days provide an external SATA port, which allows for an external SATA drive. That's option one.
USB 3.1 also provides up to 10Gbps, which is sufficient to run an external SATA III drive via USB. In fact, mechanical drive will be limited by HDD read/write speeds, not by the connection. And even an external SSD will perform barely worse on 3.1. If OP doesn't have USB 3.1 ports on the PC already, it's rather cheap to buy a card these days. I'm seeing them for $15 on Amazon. That's option two.
I still prefer internal drives for games, especially if you have an option of running an PCIe M.2 drive. But externals aren't as bad for gaming as they used to be. Latency is barely worse, and throughput is no worse than that of an internal drive.