、。
She lifted one corner of the napkin draped over the basket and gave me a quick peek.
Literal
She [topic-は] basket [on-に] is-draped napkin [genitive-の] one-corner [object-を] lift-up, briefly me [to-に] showed.
かぶせてある is the resultative ~てある: someone (presumably she) draped the napkin earlier, and the napkin remains in that draped state. Compare with かぶせている ('is in the act of draping') and かぶさっている ('is covering,' from the intransitive かぶさる) — the choice signals that a person performed an action whose result lingers. ちらっと is a 擬態語 for a quick, glancing look or a brief flash of exposure; it pairs naturally with verbs like 見る, 見せる, 覗く. The renyōkei (i-stem) form 持ち上げ used to chain into the next clause — instead of the everyday te-form 持ち上げて — gives the prose a slightly literary, narrative cadence.