You asked

What can I do if lack of sleep is hurting my relationship?

A strong relationship isn’t likely to be destroyed by lack of sleep, but being sleep deprived and grumpy isn’t likely to do it any favours either. If you’re both suffering sleep deprivation now that your baby has arrived, you’re more likely to snap at each other about trivial matters, and you’re less likely to be open to communication and compromise.
The good news is that there are several ways you can get more sleep, and that should solve any problems you may be having. One good idea is to arrange with a trusted friend or family member that they stay at your home for a night or two, and you and your partner head to a hotel. Being out of the house completely should give you both the time the rest and recharge that you need, and when you return home, you’ll both be fresh and ready to face the demands of a new baby again. Alternatively, you could have someone stay over while you’re at home, to allow you and your partner to sleep and relax, but still be around to breastfeed, and if your baby really needs you.
Once you’ve both got enough sleep to communicate calmly and rationally, sit down together and come up with long term sleep strategies that will leave you both feeling less exhausted. Alternating with baby care responsibilities, dividing up chores and lightening your workloads are all options that can work. Alternatively, you might decide to hire temporary help around the house, or with the baby.
Once you’ve come up with a strategy to get more sleep, you need to look at your sleep habits too. You may be drinking coffee too late in the day, or sleeping in a room that’s too hot. Your partner may be snoring, or there may be noise from traffic outside in your room. Make sure that your sleep habits, and those of your baby, are normal and healthy, and you should find that everyone feels happier and healthier.

More questions

Being sleep deprived does not make for a happy relationship. If you’re suffering from lack of sleep, there are a few things you can try.
It is very important that you do not use sleep remedies, including the herbal variety, before talking to your doctor - they could have an adverse affect if you are breastfeeding.
When the argument you are having involves your children, it is a much better idea to have this fight behind closed doors, away from your children.
Making sure your newborn stays healthy and happy is as much about common sense as it is about care and washing hands is one of the simplest ways to ensure it.
If you find that you and your mother are locking horns over child care issues then it’s time to assert your wishes.
Most people agree that fighting in front of children is inappropriate. However, many people also agree that, as long as your child is brought up in a loving, warm, free home, they will learn that the occasional disagreement between their loving parents is simply a part of life.
If your child is big enough you might consider a star chart for good behaviour. Each time your child goes to bed without any trouble, he or she gets a star. If at the end of the week he has seven stars on his chart then he gets a special treat.
Choosing to change child care after your parents have been looking after your child for months or years can be a delicate subject but, handled correctly, it can be done.
The right amount of sleep differs from individual to individual and even between cultures.
Negotiating the changes in friendships that a child brings can be tricky for any new mom, but it is possible.