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We prove that for every 
$m\geq 0$ there exists an 
$\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if 
$0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and 
$x$ is sufficiently large in terms of 
$m$ and 
$\unicode[STIX]{x1D706}$, then The value of 
$$\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray}$$
$\unicode[STIX]{x1D700}(m)$ and the dependence of the implicit constant on 
$\unicode[STIX]{x1D706}$ and 
$m$ may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters 
$m$ and 
$\unicode[STIX]{x1D706}$ to vary as functions of 
$x$ or replacing the set 
$\mathbb{P}$ of all primes by primes belonging to certain specific subsets.
Letting 
$p$ vary over all primes and 
$E$ vary over all elliptic curves over the finite field 
${{\mathbb{F}}_{p}}$, we study the frequency to which a given group 
$G$ arises as a group of points 
$E\left( {{\mathbb{F}}_{p}} \right)$. It is well known that the only permissible groups are of the form 
${{G}_{m,\,k}}\,:=\,\mathbb{Z}\,/m\mathbb{Z}\,\times \,\mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let 
$M\left( {{G}_{m,\,k}} \right)$ be the frequency to which the group 
${{G}_{m,\,k}}$ arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for $M\left( {{G}_{m,\,k}} \right)$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M\left( {{G}_{m,\,k}} \right)$, pointwise and on average. In particular, we show that $M\left( {{G}_{m,\,k}} \right)$ is bounded above by a constant multiple of the expected quantity when 
$m\,\le \,{{k}^{A}}$ and that the conjectured asymptotic for $M\left( {{G}_{m,\,k}} \right)$ holds for almost all groups ${{G}_{m,\,k}}$ when 
$m\,\le \,{{k}^{1/4-\in }}$. We also apply our methods to study the frequency to which a given integer 
$N$ arises as a group order 
$\#E\left( {{\mathbb{F}}_{p}} \right)$.
