i'm learning chinese as well so i don't know how good my answer will be, but from what i know it means to play and is used in like a "hanging out and having fun" sort of sense :)

In this context, 玩 means "stay at your home and kill some time." For example, watching tv, drinking tea, or simply sitting on the sofa. Yes, it changes the meaning. If you left 玩 out, it's still correct, but the person won't know what you visit for. 玩就是“打发时间”的意思 如果省略【玩】，被问者就不清楚你访问他家是为了什么。

玩 means ' hang out' in this sentence. if you want to visit formally, you can use 拜访. （我能去你家拜访吗？） you can omit both of them, if you don't emphasize the purpose.

In this Mandarin sentence, "玩" means "do something fun". So the accurate English translation is "Can I visit your home and do something fun?" it's a little bit different from "Can I visit your home?" but both sentences are correct. So you can use "玩" in your sentence or leave it out. Another option is replace it by the specific visiting reason.

Be a guest! Suitable for good friends