The Cut Clip to Clipboard button is available on the collection toolbar, except when the Clipboard tab is active. You can also find this button on the AceText Tower.
This command places the selected clips on the clipboard in the same way as the Copy Clip to Clipboard command. It then removes the selected clips from the collection. The removed clips are not added to the Recycle Bin. You can restore the cut clips using the Paste as New Clip command (before the clipboard is overwritten) or by using the Undo command (which also requires you to undo everything you did in the same collection after you cut the clips).
See Manage and Arrange Text and Look up and Reuse Text to learn how to use AceText.