Whatever the reason, it's a sad situation for both parent and child when a parent drops out of the picture in the wake of a divorce. The child, even a hurt child, may feel abandoned and lost. It's not unusual for kids who have been left to wonder if there is something essentially wrong with them that their own parent couldn't love them enough to stay. The parent loses one of the most important of human relationships. For many, the loss of their child's trust and love is the biggest failure of their life.
Yes, reconnecting can work. Having grown up or cleaned up or moved past your anger or pain from the divorce, you can re-establish or create an important role in your child's life -- even many years later. Kids of divorce generally are better off when they have the love and support of both their parents. But for it to work, you do need to do more than show up. Consider the following:
Ideally you need support for your renewal of the relationship from the parent the kids live with. Mend that relationship as best you can. Demonstrate that the children will be safe with you. You don't have to be buddies -- just civil. Make it clear that the children's well-being is what comes first, not your old differences.
If you are the one (not your child) taking the initiative, remember that you have had time to think about reconnecting. It will be a surprise to your child. Don't expect the kids to immediately respond or to respond with excitement. They are likely to be wary. They may refuse your first overture. They need time. They need a sincere apology and a simple but sincere explanation for your absence.
The kids need to get to know you to feel safe with you. Start with visits in or near their home. Keep them short. As you get more comfortable with each other, you may be able to lengthen the time together.
Take the time to get to know your kids' interests, hobbies, and style. If you can't get enough information from your ex, see about visiting the kids' school guidance counselors or teachers to get a sense of who the kids are and what they like to do. Knowing more about your children will help you plan more appropriately for your time together.
Adjust your expectations to the kids' needs. Remember: While you've been gone, they've gone on with life. They've developed friendships and ways that they spend their free time. They may be on teams or in other activities that take up after school and weekend time. It's unfair for you to ask them to adapt to your schedule. Talk with them about their schedule and work with them to find times you can spend time together without damaging their other relationships or obligations.
Clueless about how to be with a kid who is at your child's age and stage? Don't bluff it. Find someone who can give you information about what kids of that age generally like. Clueless about how to establish appropriate limits; how to be kind but also firm? Think about taking a parenting class or reading a parenting book or seeing a child therapist for pointers.
With love and patience and perhaps some guidance from the pros, you can reclaim your relationship. Just go into the project with the understanding that it will take time, probably more time than you think, for your children to trust you again.