The base of my fence seems to be popping out and I am trying to find a way to push it back in to fix it. I wish I could explain the situation better but I've attached some pics to get a better understanding. Would anyone be able to advise on how to get this fixed? Thank you in advance
This looks to be an issue caused by the elevated height of the next-door property. Have you had a word with your neighbours about it? The timber pegs that the lower retaining board were fixed to have rotted out, and now the retained soil is migrating under the fence. I don't believe you'll be able to push the board back into position until you remove the soil that has taken its place.
The removal of this soil is likely best done from the neighbour's side. If that's not an option, then you could remove the board and dig the soil out from your side. However, you might find that if the soil is not particularly stable, it might continue to migrate out as you remove what is needed. If you can remove the soil, then the board can be fixed back in place with mending plates connecting it to the front face of the post. I've added a rendering below to illustrate.