The first thing you would like to do after placing your new Personal claim is to feed it with some coins. You will have 9 tiles lands by default and they will consume some points from support pool which is zero in a fresh new monument. Miss that initial feeding and your private monument will be wiped away at midnight.
Use option "Worship" on the Monument.
Put items inside monument’s inventory window and press “Sacrifice” to add to monuments support points stock.
Monument will automatically check at in-game midnight and maintenance cost subtracted from the Stock pool.