# [05AB1E], 10 bytes

    sF³gݨm³*O

[Try it online!][TIO-j5xtm7k5]

Takes x as first argument, n as second and p in ascending order as third.

###Explanation###
    sF³gݨm³*O
    s          # Forces the top two input arguments to get pushed and swaped on the stack
                 (This is in case n is 0 so x is on the stack)
     F         # Do n times...
      ³         # Push the third input (the coefficients)
       g        # Get the length of that array...
        ݨ      # and create the range [0 ... length]
          m     # Take the result of the last iteration to these powers (it's just x for the first iteration)
           ³*   # Multiply the resuling array with the corresponding coefficients
             O  # Sum the contents of the array
               # Implicit print
[05AB1E]: https://github.com/Adriandmen/05AB1E
[TIO-j5xtm7k5]: https://tio.run/##MzBNTDJM/f@/2O3Q5vTDcw@tyD20Wcv//38jPWMuE65oXRMdXSBTx0DPMBYA "05AB1E – Try It Online"