There are many expressions with heart. One of them is to "lose your heart" to someone. When that happens, you have fallen in love. But if the person who "won your heart" does not love you, then you are sure to have a "broken heart". In your sadness, you think that the person you loved is "hard-hearted", and in fact, has a "heart of stone.
You may decide to "pour out your heart" to a friend. Telling someone about your personal problems can often make you feel better.
If your friend does not seem to understand how painful your broken heart is, you may ask her to "have a heart". You are asking your friend to show some sympathy(同情) for your situation. Your friend "has her heart in the right place" if she says she is sorry for how you feel.
If your friend says, "My heart bleeds(流血) for you", she means the opposite. She is a "cold-hearted" person and does not really care about your situation.
When you are feeling afraid, "your heart is in your mouth". You might say, for example, that your heart was in your mouth when you asked a bank to lend you some money for a new house.
If that bank says no to you, don't "lose heart". Be "strong-hearted". Sit down with the banker and have a "heart to heart" talk. Be open and honest about your situation. The bank may have a "change of heart" and agree to lend you the money. Then you can stop worrying and "put your heart at rest"