PCPs with imperfect completeness
Yes, PCPs with imperfect completeness have been studied before. The main motivation is that for some natural and interesting problems, finding whether there is a perfect solution is actually easy (polynomial-time), while approximating the best solution, if this best solution is not perfect, is (or believed to be) hard.  

Here are some examples:


The Unique Games Conjecture. Unique games with perfect completeness are easy, so the Unique Games Conjecture focuses on unique games with imperfect completeness.
Feige and Reichman showed NP-hardness for unique games for some constant completeness and soundness values (not c=1-delta s=epsilon as conjectured).
Linear equations. Linear equations with perfect completeness are easy (using Gaussian elimination), so hardness results for linear equations (e.g., Hastad's) focus on imperfect completeness.