The leaves of these lovage plants are being damaged by the larvae of the celery fly (Euleia heraclei).
The celery fly itself is tiny and mainly goes unnoticed, but once the female has laid her eggs into the leaves of your celery, or lovage, or parsnips, or parsley (or quite a few other plants), you’ll certainly notice the damage they do.
The egg is inserted under the skin of the leaf and the grub lives between the two leaf faces, chomping away and destroying the leaf from the inside. This causes the characteristic scorched appearance and blistering.
The grubs will spend two to three weeks inside the leaf, growing bigger and bigger. They will then either pupate within the leaf, or in the ground underneath the plant. A new generation of flies will emerge a few weeks later ready to start the second cycle of infestation.
The only thing you can really do, on a garden scale, if you are affected by celery flies is to pick off and destroy damaged leaves as you notice them.