Tcl, 54 bytes

proc S f {time {incr k
append s +[expr $f]} 5
expr $s}

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proc S {f s\ 0} {time {incr k
set s [expr $s+$f]} 5
set s}

If it only worked with integers, I could golf it more!

Janet, 54 bytes

|(+;(seq[k :down[5 0]](defglobal'k k)(eval-string $)))                                                                                  

Example input: (+ k (/ k 2) (math/pow k 2))

R, 35 bytes

k=1:5;sum(eval(parse(t=scan(,""))))

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TIO link includes a function solution as well (38 bytes)

Mathematica, 17 14 13 bytes

Thanks to Ian Miller for saving 3 bytes.

Thanks to LegionMammal978 for saving 1 byte.

#~NSum~{k,5}&

The input should be an actual expression containing k, e.g.:

#~NSum~{k,5}&[Sqrt[k]^3+4]

Japt, 10 bytes

6ÆK=XOxUÃx

Input string should have variable as an uppercase K. sqrt(K) should be inputted as K**0.5.

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Explanation

eval scope didn't work in my favor; had to redefine the counting variable X as a global K.

6ÆK=XOxUÃx      // implicit: U = input string
6oXYZ{K=XOxU} x // expanded

6oXYZ{      }   // create array [0, 6) and map to function:
      K=X       //   redefine the array value to global K
         OxU    //   eval the input string
              x // sum the resulting array

TI-Basic, 12 bytes

Σ(expr(Ans),K,1,5

Call with "string":prgmNAME, where string is any valid TI-Basic expression of K.

Swift, 202 184 bytes

import Foundation;func s(i:String){print([1,2,3,4,5].map{NSExpression(format:i.replacingOccurrences(of:"k",with:"\($0).0")).expressionValue(with:nil,context:nil)as!Float}.reduce(0,+))}

For some reason this will only run locally :(.

Here is an explanation of what I am doing:

import Foundation // Import the Foundation module

func s(i:String){ // Create a function that takes in a String and returns a Float

    print( // Print the result of the follow algorithm to strdout

        [1,2,3,4,5].map{ //Conduct the follow code on numbers 1 - 5

            NSExpression(format: // Create an expression with the following String and return it 

            i.replacingOccurrences(of:"k",with:"\($0).0")) // Create a string replacing all ocurrances of 'k' in `i` with the current Float from the map

           .expressionValue(with:nil,context:nil)as!Float // Get the resulting value of the expression

       }.reduce(0,+) // Add the result of all the expressions together
    )
}

Thanks to @Mr. Xcoder for saving 15 bytes!

Common Lisp, 55 bytes

(defun f(x)#.(read))(print(+(f 1)(f 2)(f 3)(f 4)(f 5)))

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Example input - output: 
(* x 2) - 30 
(sqrt x) - 8.382333 
(+ (/ x 2) x (expt x 2)) - 155/2 
(expt x 2) - 55

different, longer (58 bytes) version - starts getting shorter if you do sumation from 1 to 7.

(print #.`(+,@(mapcar #'(lambda(x)#.(read))'(1 2 3 4 5))))

yet another and longer method (65 64 bytes) - doesn't define function - just inserts your expression into a loop. Should get shorter for bigger sums.

(setf a(read)b 0)(loop as x from 1 to 5 do(incf b #.a))(print b)

dc, 31 24 bytes

?sa1k[lax+Kd1+k5>p]dspxp

The input must be given in reverse-Polish notation (also known as postfix notation) and enclosed in square brackets ([]) with:

• K replacing k as the parameter;
• + representing addition;
• - representing subtraction and _ followed by any number representing a negative number;
• * representing multiplication;
• / representing division;
• ^ representing exponentiation;
• v representing the square-root.

For example, -2*k+k+3*k**2+k**0.5-k/2 would be input as [_2K*K+K2^3*+Kv+K2/-]. This takes due advantage of the fact that K is a dc command which returns the current precision (initially set to 1). Therefore, by the end, this returns the output with a precision of 6.

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Stacked, 16 bytes

5~>[@k#~]2/"!sum

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5~> is a range from 1 to 5 incluive. 2/ makes a func dyadic, " is pair-wise, and ! is execute. This thus maps the range [1, 5] with the input, which is then evaluated after defining the range member to be k. Then, the results are summed.

Octave, 50 46 31 29 bytes

@(d)eval(["k=1:5;sum(" d 41])

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Exponentiation is denoted with the caret .^ and multiplication is denoted with .*.

This declares an anonymous function that takes in argument d. It sets k to be equal to the range 1:5 and sums the evaluated d and returns it.

APL (Dyalog), 9 bytes

+/⍎⎕⊣k←⍳5

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Addition is +, subtraction is -, multiplication is ×, division is ÷ exponentiation is * and execution is right to left, so use () to group expressions.

Input is in terms of k.

Explanation

k←⍳5                    Set k to be equal to the vector 1 2 3 4 5
⊣                       The left argument:
+/                      Sum of
⍎⎕                      The evaluated input (the eval returns an array because k is an array)

And here's a solution that takes trains as input (like the Jelly answer): +/(⍎⎕)¨⍳5.

Octave, 25 23 bytes

@(f)sum(inline(f)(1:5))

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Exponentiation is denoted as .^

Jelly, 5 bytes

vЀ5S

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Input a valid Jelly monadic chain (I golfed them down in my link).

How it works

vЀ5S
 Ѐ     for each of ...
   5        5 (implicitly converted to [1,2,3,4,5]), ...
v           evaluate the input with the above as argument
    S   and find the sum

05AB1E, 8 7 6 bytes

6G¹.VO

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Input is in postfix notation, and uses the variable N. 05AB1E is a stack-based language, so only postfix notation works.

Format of E(N): write the number(s) you want to do the operation with, and then write the sign of the operation. For example, 3+4 would be 3 4+, 3*4+2*3 would be 3 4* 2 3* +. Also note that this uses t instead of sqrt, and m instead of **, so sqrt(N) would be Nt.

Explanation:

6G¹.VO
6G     For N in range(1,6). This includes [1,2,3,4,5].
  ¹.V  Read and eval input.
     O Sum results.

Python 3, 40 37 bytes

3 bytes thanks to Arnauld.

Eval scope tricks \o/

f=lambda s,k=5:k and eval(s)+f(s,k-1)

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Uses k**0.5 instead of sqrt(k).

JavaScript (ES7), 31 30 bytes

Uses k**0.5 for sqrt(k).

f=(e,k=6)=>--k&&f(e,k)+eval(e)

console.log(f("2*k"))
console.log(f("k**0.5"))
console.log(f("k+k/2+k**2"))
console.log(f("k**2"))

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