A Watt already is “per time,” namely one Watt equals one Joule per second. “Per” means divided by, so 1W = 1J / 1s. It’s the rate of usage of energy. If you use that rate for an hour, that is a Watt *times* an hour, or a watt-hour. That’s an *amount* of energy, and in fact is equal to 3600 Joules, since there are 3600 seconds in an hour.
Think about Watts as similar to a speed. If you travel 100 kilometers per hour (100 km / h) for an hour, you go 100 km / h * 1h = 100km, since the h’s cancel out.
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