"To add that responsibility" means to take on or accept an additional task or duty.
Sentences with «to add that responsibility»
And adding the responsibility of building public support to their plate seems a bit unfair. (edutopia.org)
For much of my life, that's also how I viewed getting older: With age comes responsibility, and with added responsibility comes reduced freedom. (relevantmagazine.com)
Why add responsibility for operating a miniature power grid, too? (microgridknowledge.com)