Can you be more precise about the model? The first HTC One models were the One X, One S and One V, all of which came out in 2012. It may help to know what Android version you are running.
But I would think that the "recent tasks" screen is the thing you want. I can't recall whether the early One models have a dedicated button for that or whether you press and hold the home key, but you can switch and close tasks from that screen. It's usually best not to go around closing tasks anyway, just let the OS handle it, so task managers are not really needed most of the time.
I've not seen a HTC "active application widget", but am sure you can find such things in the Play Store if you want one.