The suggested intuition behind mixed Hodge structures - developed in particular to generalize Hodge decomposition of cohomology groups from smooth and complete Kähler manifolds to more general algebraic varieties - is that one should think the cohomology groups $H^k(X)$ to be endowed with increasing filtrations whose successive quotients originate from
cohomologies of appropriate smooth complete varieties, hence admit (pure) Hodge structures, but of different weights.

Here is an motivating and so rather 'informal' example which I took from here: Example of curves
where I missing this decisive feature that successive quotients of the associated filtration shall come from the cohomologies of smooth complete varieties.

The example works as follows:

To motivate the definition, consider the case of a reducible complex algebraic curve $X$ consisting of two nonsingular components, $X_1 $ and $X_2$, which transversally intersect at the points $Q_1$ and $Q_2$. Further, assume that the components are not compact, but can be compactified by adding the points $P_1 , ... , P_n$. The first cohomology group of the curve $X$ (with compact support) is dual to the first homology group, which is easier to visualize. There are three types of one-cycles in this group. First, there are elements $ \alpha_{i}, (i=1,..., n)$ representing small loops around the punctures $P_{i}$. Then there are elements $ \beta_{j} $ that are coming from the first homology of the compactification of each of the components. The one-cycle in $ X_{k}\subset X$ ( $ k=1,2$ ) corresponding to a cycle in the compactification of this component, is not canonical: these elements are determined modulo the span of $\alpha_{1} ,... , \alpha_{n}$. Finally, modulo the first two types, the group is generated by a combinatorial cycle $\gamma $ which goes from $ Q_{1}$ to $ Q_{2}$ along a path in one component $X_{1}$ and comes back along a path in the other component $X_{2}$. This suggests that $ H_{1}(X)$ admits an increasing filtration

$$ 0\subset W_{0}\subset W_{1}\subset W_{2}=H_{1}(X) $$

whose successive quotients $W_n/W_{n−1}$ originate from the cohomology of smooth complete varieties, hence admit (pure) Hodge structures, albeit of different weights.

The point I not understand is from (co)homology of which smooth complete variety originate the cycles $ \alpha _{i}$ generating $W_0$? Seemingly they live in $H_1(X_k)$ but it was explicitely assumed that the components $X_i$ are not compact, therefore not complete/proper in schematic setting.

On the other hand completing them to $\overline{X_k}$ means adding the points $P_i$ which would kill these non-trivial cycles. Therefore $\alpha_i$ cannot come from $H_1(\overline{X_k})$. Does anybody see how to resolve my confusion with this example?