In this grid, each across entry is linked to a down entry. The letter next to the enumeration is to link to clues and make is easier on the solver :)
The across clues all lack one letter in their wordplay that is superfluously given in the associated down clue. All of the down clues give a word, which need to have said letter removed and most need to be anagrammed to fit into the grid.
The across clues all include the definition of the linked down entry, and the down clues include the definition of the word formed by the across wordplay before the extra letter is added. These definitions must be deleted in order to cryptically parse the clues.
All enumerations are given for the actual grid entries.
Example
Across: Set down transport craft oddly (4)
The odd letters of craft are CAT, which needs an extra 'R' from the down clue. Set down is the definition of the down entry, and is ignored.
Therefore, you enter CART into the grid
Down: Blair deigned to hide feline for Scottish lord (4)
LAIRD is a hidden within b(lair d)eigned, with a def of Scottish lord. Feline is the definition of the across wordplay and is ignored. The down entry will be leftover letters (potentially anagrammed) from LAIRD (LAID), and means 'set down' (from the across clue).
Therefore you enter LAID into the grid