As the canes age, blooming tends to be on the tips. Harrison's Yellow is a once blooming rose so it sets buds in the fall. Late cutting back will be cutting off next spring's bloom.
I'd cut all the canes back to stronger wood but if you want a bloom next year, you may want to wait until then. The proper time to prune once blooming varieties is just after they bloom. Meanwhile you can tie the canes loosely together to support them until they bloom next spring.
If you selectively remove canes, just cut back the biggest and thickest ones. The stubs will send out new growth while the remaining canes will bloom next spring. After they bloom you can cut the rest of them back.