The diagram will show how this puzzle is to be solved. It is the only way within the conditions laid down. 
Starting at the pudding with holly at the top left-hand corner, we strike out all the puddings in twenty-one straight strokes, taste the steaming hot pudding at the end of the tenth stroke, and end at the second sprig of holly.

Here we have an example of a chess rook's path that is not re-entrant, but between two squares that are at the greatest possible distance from one another. 
For if it were desired to move, under the condition of visiting every square once and once only, from one corner square to the other corner square on the same diagonal, the feat is impossible.

There are a good many different routes for passing from one sprig of holly to the other in the smallest possible number of moves—twenty-one—but I have not counted them. 
I have recorded fourteen of these, and possibly there are more. 
Any one of these would serve our purpose, except for the condition that the tenth stroke shall end at the steaming hot pudding. 
This was introduced to stop a plurality of solutions—called by the maker of chess problems "cooks." 
I am not aware of more than one solution to this puzzle; but as I may not have recorded all the tours, I cannot make a positive statement on the point at the time of writing.