"To have some responsibility" means that you are accountable for a certain task or job. It means you have a duty or obligation to take certain actions or make sure something gets done. Full definition
Meanwhile, the male who is her lover doesn't not have the responsibility for contraception, but he try to push out of his responsibility. (globaleducationmagazine.com)
We also have the responsibility of making our foster home a mud platter, as not to make the dogs adoptive home seem like less of a home than ours. (bamabully.org)
Maybe they also have some responsibility for what happens when their children arrive home from school. (articles.chicagotribune.com)