When the chain is on the smallest chainring is there any slack in the cable? There should not be. Slack can be removed at the cable fixing screw on the derailleur. If you try removing slack here be careful to reattach the cable with the correct cable routing.
You should be able to add cable tension with a barrel adjuster at the end, sometimes in the middle, of the cable housing. Usually a counter- clockwise turn of the barrel adjuster lengthens the housing, adding tension by forcing the cable to take a slightly longer route.
It should not take much of a change.