River Clip

The River Clip is used to enter Hyrule Castle or Agahnim's Tower with the help of Major Glitches

With Boots and the Hookshot

Position Link along the slope on the y-coordinate range 5B4-5B8. If you're higher than this, Link will dash along the slope rather than jump. If you're lower than this, you will jump at the wrong angle.
Do an Item Dash with the Hookshot (Y+A) and move to the right while holding B
If you can not move any further to the right, hold A and you automatically drop down the ledge
Dash 1 screen to the right, move a bit down, and continue dashing to the right. The bush is a useful cue for when to cancel your dash and move down.
You will enter the screen west of Hyrule Castle and can now enter Hyrule Castle or Agahnim's Tower.

River Clip with Boots

Position Link along the slope on the y-coordinate range 5B4-5B8. If you're higher than this, Link will dash along the slope rather than jump. If you're lower than this, you will jump at the wrong angle.
Dash Buffer Clip right to at least x-coordinate 5C9. Further right will work as well but this is the earliest you can jump for all of the y-coordinate positions.
Hold A to charge a dash and jump downwards.
Dash 1 screen to the right, move a bit down, and continue dashing to the right. The bush is a useful cue for when to cancel your dash and move down.
This video shows the East Hyrule Castle Entrance

If you bonk while having EG, you have to reset your console to get back to the state. Depending on the category, the run is over
If you are fast enough, you do not need to kill the Green Guardian

With Boots