Suppose $X\subset \mathbb{CP}^N$ is a $n$ dimensional projective manifold (and complex dimension $n>1$), take a general projection $p\colon X\to\mathbb{CP}^{n+1}$. Suppose $H_1(X)$ is nontravial. Does it hold $H_1(p(X))$ is nontrivial?
(It would show the smoothness condition on the ample divisor in Lefschetz hyperplane theorem is necessary)
No, that is not true, and you can make counterexamples by considering projections of projective bundles over a curve. Let $C$ be a smooth, projective curve. Let $n\geq 5$ be an integer. Let $X$ be $C\times \mathbb{P}^{n-1}$. Let $\mathcal{L}_C$ be a very ample invertible sheaf on $C$, and denote $H^0(C,\mathcal{L}_C)$ by $V_C$. For every closed point $t$ in $C$, there is an associated surjective linear transformation, $$\text{ev}_t:V_C \to \mathcal{L}_{C,t},$$ where $\mathcal{L}_{C,t}$ is the $1$-dimensional vector space, $$\mathcal{L}_{C,t} = \mathcal{L}\otimes_{\mathcal{O}_C}\mathcal{O}_{C,t}/\mathfrak{m}_{C,t}.$$ Since $\mathcal{L}$ is very ample, for every pair $s$, $t$ of distinct closed points of $C$, the following associated linear transformation is surjective,$$ (\text{ev}_s,\text{ev}_t):V_C \to \mathcal{L}_{C,s}\oplus \mathcal{L}_{C,t}. $$
Denote $H^0(\mathbb{P}^{n-1},\mathcal{O}_{\mathbb{P}^{n-1}}(1))$ by $V_{\mathbb{P}^{n-1}}$. Denote by $\mathcal{L}_X$ the invertible sheaf on $X$, $$\mathcal{L}_X = \text{pr}_C^*\mathcal{L}_C\otimes_{\mathcal{O}_X} \text{pr}_{\mathbb{P}^{n-1}}\mathcal{O}_{\mathbb{P}^{n-1}}(1).$$ Denote by $V_X$ the vector space of global sections, $$ V_X = H^0(X,\mathcal{L}_X) = H^0(C,\mathcal{L}_C)\otimes_{\mathbb{C}}H^0(\mathbb{P}^{n-1},\mathcal{O}_{\mathbb{P}^{n-1}}(1)) = V_C\otimes_{\mathbb{C}}V_{\mathbb{P}^{n-1}}.$$ For every closed point $t$ of $C$, since $\text{ev}_t$ is surjective, also the following linear transformation is surjective, $$\text{ev}_{X,t}: V_X = V_C\otimes_{\mathbb{C}}V_{\mathbb{P}^{n-1}} \to \mathcal{L}_{C,t}\otimes_{\mathbb{C}}V_{\mathbb{P}^{n-1}}.$$ It follows that $\mathcal{L}_X$ is globally generated and there is an induced morphism, $$\phi:X\to \mathbb{P}(V_X^\vee).$$ Moreover, $\mathcal{L}$ is very ample, and for every pair $s$, $t$ of closed points of $C$, since $(\text{ev}_s,\text{ev}_t)$ is surjective, also the following linear transformation is surjective, $$ \text{ev}_{X,s,t} : V_X = V_C\otimes_{\mathbb{C}}V_{\mathbb{P}^{n-1}} \to (\mathcal{L}_{C,s}\oplus \mathcal{L}_{C,t})\otimes_{\mathbb{C}} V_{\mathbb{P}^{n-1}}.$$ It follows that $\mathcal{L}_X$ is very ample and $\phi$ is a closed immersion.
Let $W\subset V_X$ be a linear subspace of dimension $n+2$ parameterized by a point of the complex Grassmannian, $G= \textbf{Grass}(n+2,V_X)$. There is an associated linear projection, $$ p_W : \mathbb{P}(V_X^\vee)\setminus \mathbb{P}(W^\perp) \to \mathbb{P}(W^\vee). $$ For every closed point $t$ in $C$, since $\text{ev}_{X,t}$ is surjective, there is a Schubert cycle $\Sigma_t$ of codimension $3$ in $G$ parameterizing $W$ such that the codimension $n+2$ subspace $\mathbb{P}(W^\perp)$ has nonempty intersection with the dimension $n-1$ subspace $\phi(\{t\}\times \mathbb{P}^{n-1})$. There is a subvariety $\Sigma_C$ of $G$ with codimension $2$ that contains every $\Sigma_t$.
Similarly, for every pair of distinct closed points $s$, $t$ of $C$, there is a Schubert subvariety $\Sigma_{s,t}$ of $G$ with codimension $n+1$ parameterizing $W$ such that the induced linear transformation $$ W\hookrightarrow V_X \xrightarrow{\text{ev}_{X,s,t}} (\mathcal{L}_{C,s}\oplus \mathcal{L}_{C,t})\otimes_{\mathbb{C}}V_{\mathbb{P}^{n-1}} $$ has rank $\leq n+1$. There is a subvariety $\Sigma_{C\times C}$ of $G$ with codimension $n$ that contains every $\Sigma_{s,t}$. Since $n\geq 1$, there exists $W$ such that the corresponding point of $G$ is not in the closed subset $\Sigma_C$ of codimension $2$, nor is it in the closed subset $\Sigma_{C\times C}$ of codimension $n$.
For such $W$, the closed subset $\mathbb{P}(W^\perp)$ is disjoint from $\phi(X)$. Moreover, for fixed closed point $s$ of $C$, there is a projective subbundle $Y_s \subset X$ of codimension $2$ such that $p_W(\phi(Y_s))$ is contained in the $\mathbb{P}^{n-1}$, $p_W(\phi(\{s\}\times \mathbb{P}^{n-1}))$. Thus, there is a commutative diagram of pushforward maps of fundamental groups $$\begin{array}{ccc} \pi_1(Y_s) & \to & \pi_1(p_W(\{s\}\times \mathbb{P}^{n-1})) \\ \downarrow & & \downarrow \\ \pi_1(X) & \to & \pi_1(p_W(X)) \end{array}$$ Since the first vertical arrow is an isomorphism, and since $\pi_1(\mathbb{P}^{n-1})$ is trivial, it follows that the bottom horizontal arrow is trivial.
That does not immediately imply that $\pi_1(p_W(X))$ is trivial. Denote by $U\subset X$ the maximal open subset such that $p_W:U\to p_W(U)$ is an isomorphism. Let $x$ be a closed point of $U$. Let $g:Z\to p_W(X)$ be an covering space. There is a unique structure of complex analytic variety on $Z$ such that $g$ is holomorphic. Let $z$ be a closed point of $Z$ with $g(z) = x$. By the previous paragraph, there is a unique holomorphic morphism $h:X\to Z$ such that $g\circ h$ equals $p_W$ and such that $h(x)$ equals $z$. So the image of $h$ is an irreducible component of $Z$. Of course if $h(X)$ projects isomorphically to $p_W(X)$, then $g$ is a trivial cover. Thus assume that $h(X)$ does not project isomorphically to $p_W(X)$.
Since, locally near a general point $p_W(y)$ of $p_W(X\setminus U)$, $p_W(X)$ has two branches, also locally near $h(y)$, $Z$ has two branches. Using connectedness of the double locus in $p_W(X)$ (every codimension $2$ subvariety of $\mathbb{P}^{n+1}$ is connected if $n\geq 3$), it follows that $Z$ has precisely two irreducible components. However, since $n\geq 5$, there are also triple points of $p_W(X)$. This contradiction implies that $g$ is an isomorphism. Therefore $p_W(X)$ is (topologically) simply connected. Therefore also $H_1(p_W(X);A)$ is zero for every Abelian group $A$.