Departure Controller reported that when he was distracted from the scope by having to read a clearance; the aircraft received an RA with another aircraft he was working. Reporter cited Tower failure to issue correct clearance as contributory.

Narrative

A Citation X responded to a TCAS RA with a T28. The T28 [was a] VFR overflight at 5;500 transitioning from northeast to southwest. The Citation X; IFR; [was] a departure climbing to 5;000. The Citation X departed on a 340 heading on another departure sector tag. I radar identified the aircraft and climbed [him] to 5;000. The flight data block scratch pad information indicated the aircraft should be routed over a fix not in my sector. I asked the pilot what the first fix was on his route of flight. The pilot responded with XXXXX..JXXX. This is not what the data block or flight progress strip indicated. I then had flight data full route the Citation X to see if multiple flight plans were stored. There was only one; routing the aircraft over the intersection and with FRC (full route clearance) to ZZZ in the remarks section. I then advised the pilot he would need to be re-routed. I issued the correct route to the pilot. The pilot advised he missed most of the clearance due to having to responded to a TCAS-RA. I did not notice the proximity of the Citation X and the T28 due to my eyes being down reading the clearance. The Citation X never descended out of 5;000; I believe he just slowed his rate of climb. No traffic was exchanged; when I looked up after reading the clearance the aircraft had already passed each other. Have the airport flight data issue the correct clearance.