To determine which of the given expressions represents a factorization from left to right, let's examine each option step by step:

**Option A: $2a^{2}-b^{2}=\left(a+b\right)\left(a-b\right)+a^{2}$**

- The expression on the right side, $\left(a+b\right)\left(a-b\right)+a^{2}$, does not simplify to the left side, $2a^{2}-b^{2}$, because $\left(a+b\right)\left(a-b\right) = a^2 - b^2$. Adding $a^2$ to this would result in $2a^2 - b^2 + a^2$, which is not the original expression. Therefore, this is not a correct factorization.

**Option B: $2a\left(b+c\right)=2ab+2ac$**

- This expression represents the distributive property, where $2a$ is multiplied by both $b$ and $c$, resulting in $2ab + 2ac$. This is a correct application of polynomial multiplication, but it is not a factorization process. Therefore, this option does not represent a factorization.

**Option C: $x^{3}-2x^{2}+x=x\left(x-1\right)^{2}$**

- To verify this, let's expand the right side:
$$x(x-1)^2 = x(x^2 - 2x + 1) = x^3 - 2x^2 + x$$
This matches the original expression on the left side, indicating that this is indeed a correct factorization.

**Option D: $x^{2}+x=x^{2}\left(1+\frac{1}{x}\right)$**

- Simplifying the right side gives $x^2 + x^2\left(\frac{1}{x}\right) = x^2 + x$, which matches the left side. However, the expression involves division by $x$, which is not a typical form of polynomial factorization. Therefore, this option does not represent a standard factorization.

Based on the analysis, the correct answer is:

$$\boxed{C}$$