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Let q=pm where p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from Fq to Fp and zetap=e2pii/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: SigmaxisinF qchi(alphaxp k +1+betax2) (alpha, betaisinFq) where chi(x)=zetap Tr(x) is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1 and C2 over Fpt with parity-check polynomial h2(x)h3(x) and h1(x)h2(x)h3(x), respectively, where t is a divisor of d=gcd(m, k), and h1(x), h2(x) , and h3(x) are the minimal polynomials of pi-1, pi-2, and pi-(p k +1) over Fpt, respectively, for a primitive element pi of Fq. 2) We determine the correlation distribution between two m-sequences of period q-1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008).