As pointed out by Alex in his comment, this is in general not true.
For instance, consider the case $N=d=n=m=2$. Then $|L|$ is the linear system of plane curves of degree $2$ passing through $p_1$ and $p_2$ with multiplicity $2$. Of course there is only one such a curve, namely twice the line through $p_1$ and $p_2$. In particular, $|L|$ is not empty but no element in $|L|$ has ordinary singularities.

However, what you want holds under some additional hypotheses. For instance, the following result is proven in Shustin's paper *Lower deformation of isolated hypersurface singularities*, Proposition 10 p. 242.

For any pair $N$, $k$ of positive integers set $$M(N, \,k):=2 \cdot \binom{k+N}{N}, \quad M(N, \,2k-1):=\binom{k+N}{N} + \binom{k+N-1}{N}.$$
Then the following holds.

**Proposition.** Assume $$\sum_{i=1}^n \binom{m_i+N-1}{N} < M(N, \, d).$$ Then if $p_1, \ldots, p_n$ are *general* points in $\mathbb{P}^N$ there exists a hypersurface $L$ of degree $d$ having ordinary singularities of multiplicity $m_1, \ldots, m_n$ at the points $p_1, \ldots, p_n$ and which is smooth elsewhere. Consequently, the general element of $|L|$ has the same properties.

notwhat you want! – Alex Degtyarev Mar 15 at 23:40