The concept that carrying more weight will require more power being applied may be an over simplification. Yes, adding the proportionate increase in power will keep everything else more or less in line, but it's not the only way.
For a certified aircraft, the handling and performance information you are provided, will be applicable at the maximum permitted weight, unless it says otherwise. Of course, in most cases, the performance charts allow the pilot to select information with weight as a factor. But, if in doubt, keep it within the weight and C of G limits, and you'll be fine.
The maximum certified weight will be based on one or both of structural capacity or performance. An efficiently designed aircraft will limit at both very close to each other. Or else, you have more of one of those than you really need.
For aircraft where performance was in excess, sometimes a small structural improvement will allow a higher gross weight - a C 182 can have a higher gross weight when an STC'd wingstrut change is installed.
For aircraft with excess structure, like the venerable C 150, if you increase performance (with.... yes, more power) a heavier gross weight is possible - hence the C 152 with 70 pound heavier gross weight, because it's about 10 HP more powerful.
But otherwise, all other things being equal, if you want to carry more weight in a power plane, some more power will be your best solution - more fuel burn...