I'm a little confused and can't find what I need to do.

So we bought this new house. It has some grass, although partly in shade. (clay ground)
It also has a lot of roots in it. I started to pull some out as the lawn mover hates hitting against it. Bit there are so many...
I just wonder, should I pull them all out by hand. Or is there a "trick" or "short cut" I'm missing?

See pics.

Any help would be greatly appreciated!

These are the trees in my backyard... wonder what tree the roots are from

Hope someone can help me!

I have the same problem but i know what tree they are coming from.Mine are 2 to 3 inches in diameter so i'm taking a digging iron and getting under the root with a block of wood and take my ax and start cutting them out.I know it's easier said than done.Good luck.

Try looking at where the roots are aligned with a tree.
Go as far to the tree as possible, then use a very sharp handsaw, and
cut down into the turf, at right angle to the root, at some point you will meet the root, cut through, if it's not to deep to manage, dig down to expose the cut, and maybe rope the end and pull it out with some form
Of traction.
Then make good the ripped area, back to lawn.
I have done this with success on more than one occasions, just don't bite off more than you can chew.
Good luck witch ever way you go.
John.