To simplify the expression

\[
\frac{x}{x^2 - 5x + 6} - \frac{2}{2 - x} + \frac{x}{x - 3},
\]

we first factor the quadratic expression in the denominator of the first fraction.

### Step 1: Factor \(x^2 - 5x + 6\)
The polynomial \(x^2 - 5x + 6\) can be factored as:

\[
x^2 - 5x + 6 = (x - 2)(x - 3).
\]

### Step 2: Rewrite the expression
Substituting this back into the original expression gives us:

\[
\frac{x}{(x - 2)(x - 3)} - \frac{2}{2 - x} + \frac{x}{x - 3}.
\]

### Step 3: Simplify \(\frac{2}{2 - x}\)
Notice that \(2 - x = -(x - 2)\), so we can rewrite the second fraction:

\[
\frac{2}{2 - x} = -\frac{2}{x - 2}.
\]

### Step 4: Rewrite the entire expression
The expression now looks like this:

\[
\frac{x}{(x - 2)(x - 3)} - \left(-\frac{2}{x - 2}\right) + \frac{x}{x - 3} = \frac{x}{(x - 2)(x - 3)} + \frac{2}{x - 2} + \frac{x}{x - 3}.
\]

### Step 5: Find a common denominator
The common denominator for the entire expression will be \((x - 2)(x - 3)\). We will rewrite each term with this common denominator:

1. The first term is already over the common denominator:

\[
\frac{x}{(x - 2)(x - 3)}.
\]

2. The second term \(\frac{2}{x - 2}\) can be rewritten as:

\[
\frac{2(x - 3)}{(x - 2)(x - 3)}.
\]

3. For the third term \(\frac{x}{x - 3}\), we rewrite it as:

\[
\frac{x(x - 2)}{(x - 2)(x - 3)}.
\]

### Step 6: Combine the fractions
Now we can combine all the terms:

\[
\frac{x + 2(x - 3) + x(x - 2)}{(x - 2)(x - 3)}.
\]

Simplifying the numerator:

\[
x + 2(x - 3) + x(x - 2) = x + 2x - 6 + x^2 - 2x = x^2 + x - 6.
\]

So the expression can be rewritten as:

\[
\frac{x^2 + x - 6}{(x - 2)(x - 3)}.
\]

### Step 7: Factor the numerator
Factoring \(x^2 + x - 6\):

\[
x^2 + x - 6 = (x - 2)(x + 3).
\]

### Final step: Rewrite the complete expression
This gives us:

\[
\frac{(x - 2)(x + 3)}{(x - 2)(x - 3)}.
\]

### Step 8: Cancel out the common factors
Assuming \(x \neq 2\), we can cancel \(x - 2\):

\[
\frac{x + 3}{x - 3}.
\]

### Final Result
Thus, the simplified form of the expression is:

\[
\frac{x + 3}{x - 3} \quad \text{for } x \neq 2, x \neq 3.
\]