The most common and most likely answer to this crossword clue is the 5 letter word CHASE
C
H
A
S
E
More clues leading to the result CHASE
- Pursue ornament with engraving
- Pursue legal action around hospital
- Run after husband when entering church
- Piece of music has encore to follow
- Caught, possesses ecstasy, being run after
- Quarry gets in civil engineer
- Maybe R after HR?
- Court has to be held in church
- Follow lawsuit involving hospital
- Leading counsel hopeful as suit enters court
Buddy explains!

The straight part
We know that "run after" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case run after = chase.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CHASE mean?
CHASE noun- The act of one who chases another; a pursuit.
- A hunt.
- A children's game where one player chases another.
- A large country estate where game may be shot or hunted.
- Anything being chased, especially a vessel in time of war.
- A wild animal that is hunted.
- Any of the guns that fire directly ahead or astern; either a bow chase or stern chase.
- (real tennis) The occurrence of a second bounce by the ball in certain areas of the court, giving the server the chance, later in the game, to "play off" the chase from the receiving end and possibly win the point.
- (real tennis) A division of the floor of a gallery, marked by a figure or otherwise; the spot where a ball falls, and between which and the dedans the adversary must drive the ball in order to gain a point.
- One or more riders who are ahead of the peloton and trying to join the race or stage leaders.
- To pursue.
- To consume another beverage immediately after drinking hard liquor, typically something better tasting or less harsh such as soda or beer; to use a drink as a chaser
- To attempt to win by scoring the required number of runs in the final innings.
- To swing at a pitch outside of the strike zone, typically an outside pitch
- To produce enough offense to cause the pitcher to be removed
- A rectangular steel or iron frame into which pages or columns of type are locked for printing or plate-making.
- A groove cut in an object; a slot: the chase for the quarrel on a crossbow.
- A trench or channel or other encasement structure for encasing (archaically spelled enchasing) drainpipes or wiring; a hollow space in the wall of a building encasing ventilation ducts, chimney flues, wires, cables or plumbing.
- The part of a gun in front of the trunnions.
- The cavity of a mold.
- A kind of joint by which an overlap joint is changed to a flush joint by means of a gradually deepening rabbet, as at the ends of clinker-built boats.
- To groove; indent.
- To place piping or wiring in a groove encased within a wall or floor, or in a hidden space encased by a wall.
- To cut (the thread of a screw).
- To decorate (metal) by engraving or embossing.