Have out
If you have something out, like a tooth or an organ, it's removed from your body.
have round 
If you have people round, they visit your home, usually for a meal or for a social gathering.
Head for
to go in a certain direction or towards a particular place or situation
Head off (1) INFORMAL
to leave a place
head off (2)
to stop something bad from happening
Head up
to act as the head of a group or team, or to manage an organization
Hear about
If you hear about something, someone tells you about it or you find out about it from an information source like the news media.
Hear from
If you hear from someone, they call you, email you or contact you in some other way.
Hear of
If you've heard of someone, you know who they are and what they do.
Hear out
If you hear somebody out, you listen to them until they've finished what they are saying.
heat up (1)
If you heat something up, you make it hotter.
heat up (2)
If something like a discussion, a contest or a conflict heats up, it gets more intense, more exciting or more dangerous.
Help out
If you help someone out, you do something to help them.
Hem in
to restrict someone's movement, or to limit someone's freedom
Hide away
If you hide away, you go to a place where very few people can find you.
Hit back
If someone is beating or attacking you and you hit back, you fight to defend yourself.
Hit on
to have a idea or to think of something
Hold against
to dislike somebody, or be angry with them, because you blame them for something bad that happened in the past