i was a single mum to my daughter from her being four.
because your whole life is invested in raising the child, everything they think, say, do really matters to you. step back.
this boy, 16, doesn't want you to have a relationship with a man. well, he's old enough to get over it. start looking for someone.
he wants to go to his dad's. let him. let him know that you will be working on building up your own independent life while he's away. you must do this, or your life will be empty and you will sit by the phone waiting for his call until you eventually die of old age and loneliness.
the fact that its his dad, and dad can offer more material benefits, is a red herring. if it wasn't dad, it would be something else. you aren't inadequate, he's just growing up.
my daughter was rarely at home from being sixteen. she went on courses, she went to visit friends. she established a life of her own. i wasn't so good at that.
your boy is the right age for branching out alone. you've done a great job, you've brought him up well. now teach him how to pack a bag (he's a boy, it won't come naturally) and practise your goodbye wave.
and start planning your own life. work? training? clubs/activities? reading, long soaks in the bath? whatever you like that doesn't tie you down or make you miserable. get out there and smile at people.
you aren't redundant as a mum, you've just moved into a new phase. enjoy.