Large cargo shortages or other substantial cargo-related claims are settled with the cargo insurers. Insurers first indemnify the cargo-owners for the damage and then, by way of subrogation, sue the carrier in the name of the cargo-owners if, of course, there is an actionable fault of the shipowner.
Per Grose J in Ritchie v Atkinson (1808) 10 East 295 at p.309:
… to construe the delivery of a complete cargo to be a condition precedent to the right to recover any freight would be manifestly unjust; because if default were made in ever so small a proportion of that which would be a complete cargo, the master would not be entitled to any freight for however large a proportion of goods he may have delivered. The reasonable construction therefore is, that the plaintiff should receive freight according to the quantity per ton which he has delivered.
Per Willes CJ in Dakin v Oxley (1864) 15 CBNS 646 at p.667:
In both classes of cases, whether of loss of quantity or change in quality, the proper course seems to be the same, viz. to ascertain from the terms of the contract, construed by mercantile usage, if any, what was the thing for the carriage of which freight was to be paid, and by the aid of a jury to determine whether that thing, or any and how much of it, has substantially arrived. …If it has arrived, though damaged, the freight is payable by the ordinary terms of the charterparty j and the question of fortuitous damage must be settled with the underwriters, and that of culpable damage in a distinct proceeding for such damage against the ship captain or owners.