President Biden announced on Wednesday the cancellation of $7.7 billion in student loans held by 160,000 borrowers, building on his strategy of chipping away at college debt by tweaking existing programs as his administration pursues a larger forgiveness plan.

Many borrowers in this round — who qualified through public service loan forgiveness, the president’s SAVE plan or another income-driven repayment plan — have already begun receiving emails notifying them of their approvals, the…