In American football, a Hail Mary pass is a long, desperate pass at the end of the game that is hoped may gain some points, so it is used for a desperate attempt to resolve a serious problem at the last minute.
If someone has a hair of the dog, they have an alcoholic drink as a way of getting rid of a hangover, the unpleasant effects of having drunk too much alcohol the night before. It is commonly used as a way of excusing having a drink early on in the day.
Hand in hand= work together closely When people in a group, say in an office or in a project, work together with mutual understanding to achieve the target, we say they work hand in hand. There is no lack of co-operation and each synchoranises the activity with that of the other.
Women have a great power and influence because they have the greatest influence over the development of children- the hand that rocks the cradle. ('The hand that rocks the cradle rules the world' is the full form.)
A hangdog expression is one where the person's showing their emotions very clearly, maybe a little too clearly for your liking. It's that mixture of misery and self-pity that is similar to a dog when it's trying to get something it wants but daren't take without permission.
If you have a crack at something, you try to do it. If someone is attempting to do something and they are unsuccessful, you might say, "Let me have a crack at it" suggesting that you might be successful at performing the task. ('Take a crack' is also used.)