$\newcommand{\la}{\lambda}
$
Let us formalize the question as follows: 

>Take any complex number $\la$. Let $K(s,t):=s\wedge t-st$ for $s,t$ in $[0,1]$. Find $x\in L^2[0,1]$ such that 
\begin{equation*}
	x(t)-\la \int_0^1 K(s,t)x(s)\,ds=t \tag{1}
\end{equation*}
for almost all $t\in[0,1]$. 

Note that the kernel $K$ is the covariance function of the Brownian bridge, with the eigenfunctions $\sin j\pi\cdot$ and the corresponding eigenvalues $1/(j^2\pi^2)$ for natural $j$; see e.g. Section [The Brownian bridge][1]. Also, for $t\in[0,1)$ 
\begin{equation*}
	t=\sum_{j=1}^\infty b_j\sin j\pi t,\quad\text{where}\quad b_j:=\frac{2(-1)^{j-1}}{j\pi}. 
\end{equation*}
Therefore, equation (1) can be rewritten as 
\begin{equation*}
x(t)=\sum_{j=1}^\infty c_j\sin j\pi t \tag{2}	
\end{equation*}
with 
\begin{equation*}
	c_j-\frac\la{j^2\pi^2}\,c_j=b_j, 
\end{equation*}
so that 
\begin{equation*}
c_j=\frac{b_j}{1-\la/(j^2\pi^2)}; \tag{3}
\end{equation*}
this unique solution $x$ defined by (2)--(3) exists iff $\la\notin\{j^2\pi^2\colon j=1,2,\dots\}$. 

[1]: https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem