A "chain gobbler" is a recumbent device that takes up slack in a chain.
A regular bike derailleur gear system can do this with the cage in the rear derailleur, and they may have a capacity of perhaps 30 links total, but that also has to account for the gear combinations between big-big, and small-small.
A recumbent can send the chain completely back on itself in the area of the boom, something like this:
In this image is a common style, specifically a Terratrike part but most are similar. The pedals are to the right, the head tube is to the left. The yellow is the fixed boom and the smaller black tube is the inner boom.
The top straight run of chain is the power line that is under full tension from pedals and chainring, and the return line below is under far less tension. The lower "gobbler jockey wheel" is attached to the fixed frame and the other one is attached to the moving boom either permanently or by a clamp that can be repositioned
As you slide the boom in or out, the chain is "gobbled" by this extra zig-zag. Without this, the short-legged rider will have slack chain which can flap around and catch the wheel, pedal, ankle, or similar.
My understanding is that the gobbler is frequently either spring-loaded to help maintain chain tension, OR the top pulley's position can be set such that the rear derailler still adds enough tension.
Main downsides are added weight and losses in power - you're turning two extra wheels and even with larger 608 bearings inside there are still losses. Noise levels may increase slightly too.
As for extending the chain, depending on the wear it might be time for an all-new chain. Mine both take three separate road bike chains to get sufficient length. You should inspect the chain for wear, as well as the jockey wheels and the cassette. If your chain is already quite new, (under 200 km riding) then scabbing in an extra length of chain is probably fine, however the difference between the old/worn and the new section of chain means the chainrings and cassette will wear faster. Its really your call which you choose.