So, there are some studies on this, but just as HR doesn’t have an exact relationship to cycling power, HR doesn’t have an exact relationship to running power.
TL;DR: there is a relationship, but it’s not exact. That said, there are rules of thumb that we can use.
Some running studies look at energy expenditure using the exhaled gas method: just as in cycling, you get get hooked up to a metabolic cart that measures how much CO2 you exhale. Then, since we have a pretty reasonable idea of the Krebs cycle and how energy is produced, you can calculate how much energy was expended to run at a particular speed on a treadmill. These studies show that there’s a range across individuals in their energy expenditure, but that range usually falls into something like 0.8 - 1.1 kcal/kg/km.
Thus, there’s a fairly useful “rule of thumb” which, although imprecise for any individual, is generally okay for groups of people: at submaximal running paces on flat surfaces, energy expenditure is around 1 kcal/kg/km. Importantly, this is the rule of thumb you’ll find in many exercise physiology textbooks, like McArdle, Katch, and Katch.
So, on a firm flat surface, you can use 1 kcal/kg/km. That means if you weigh 75 kg, a rule of thumb is that you’d be expending 75 kcal/km. Notice it’s not dependent on running speed (that’s a clue that this is just for aerobic running, not sprinting), so if you run a km in 5 minutes (=300 seconds), that’s 75 kcal/300 seconds, or 250 calories per second.
We know how to convert a calorie to a joule; there are about 4.18 calories per joule. But, here’s the key: human gross metabolic efficiency is around 20-25%. You may know this. If we assume GME is 23.9%, then we have another handy rule of thumb: that 1 kcal is “equivalent” to 1 kilojoule.
So work this backward, and it turns out we can get a third rule of thumb: running speed in m/s is approximately equal to watts/kg.
As a rough check, notice that with this rule of thumb, running a marathon at around 2h would be just under 6 watts/kg. So, a world-class marathoner would need to produce about the same watts/kg as a world-class cyclist. Anecdotally, people who both run and cycle say that running a 10k in 42:00 is roughly about as hard as breaking an hour for a 40k TT. Using the rule of thumb, a 42:00 10k would require about 4 watts/kg. So that’s in the right ballpark, too.
That’s for running on flat ground. We know how to calculate the energy required if you had to climb (or descend) a hill, too, so that gets added to the rule of thumb above. This is how Phil Skiba estimates running power from speed and elevation change.
Obviously, if your son is a gifted runner, his energy expenditure may be less than 1 kcal/kg/km, and if he’s more efficient than 23.9%, that will also change the exact conversion–but you can see above how and where the calculation would change.