We often use "give away" when we say a person gives something to many as a gift, rather than just one person.
For example, the teacher gave away candies to his students.
Can we say "the teacher gave away the candy to Mary" (just 1 person, not many people)?
Short answer: no. It doesn't sound right.
Other thoughts: If you keep the structure but switch the word order. Changing your last sentence to this: 'the teacher gave the candy away to Mark', it sounds a little better but still it misses the mark.
Even with multiple subjects the away part feels off. For the phrase 'give-away', it makes sense to keep the structure when using as a noun but when using as a verb, drop the away.
To give away has many definitions. In your example it means to get rid of or donate something. When you give somethiing away, however, you generally don't know who the recipient is, or the recipient is anonymous, or it might be an organization that distributes what you give them to another set of unknown people. If you know who's getting it, you typically don't use "give away." For example:
I'm giving away all my possessions.
I'm givingawayall my possessions to my brother.
In the next example, you can use away in the second sentence because the recipients are still anonymous. You would not typically use it in the third sentence.
I'm giving away 10 prizes.
I'm giving [away] 10 prizes to the first 10 people who come to the exhibit.
I'm givingawaythe best prize to Tom because he worked the hardest.
There are many exceptions to this and examples we could all invent that use give away with known recipients. But if you follow this, you'll sound like a native speaker more often than if you don't.
Also keep in mind that there are several other definitions of to give away. There's also a giveaway, which is typically something inexpensive that you give to a large group of people.
