dnd 5e: Can I do several classes and cast a spell that I already know as a different class?

The section in the PHB on spell casting while multiclasses will help with much of its confusion.

When multiclasses as a spellcaster, the rules change slightly with respect to how your spells are prepared / known and cast.

Every spell you know and prepare is associated with one of your
classes, and you use the ability to cast spells of that class when
cast the spell

Let us assume that you only know to heal wounds from your sorcerer class and you never learned it from bard. In this case, when you throw to heal wounds It is always considered cast as a sorcerer's spell. What this means is that it will use your sorcerer's spell casting modifier and save DC spells and any other quality that affects the spell you are casting. It would also qualify for any feature that depended on something being a sorcerer's spell but could not take advantage of anything that required it as a bard spell.

However, you share Pact Magic slots and slots between classes

A place where spells overlap is in which spaces you can use to cast them.

If you have both the spell casting class function and the pact magic class function of the sorcerer class, you can use the spell slots you get from the pact magic function to cast spells you know or have prepared from classes with the spell casting class function, and you can use the spell slots you get from the spell casting class function to cast sorcerer spells you know.

So you can throw to heal wounds (still as a sorcerer) but using whatever slot machines he has available. For example, like Warlock 1 / Bard 3, you would have 5 first level spaces and 2 second level spaces available to cast any spell you know. Which means you could throw that to heal wounds using a second level slot if you wanted.