Interpretation of p-value near alpha level If the alpha is set to 0.05 I have encountered many scientific publications saying that a "tendency for an effect" is present based on a p-value of 0.05 < p-value < 0.1. On the other hand, I have had statisticians who criticize me for doing so, because there is only "reject" or "not reject". Therefore, it does not make sense to distinguish between a p-value of 0.08 or 0.97. Also, some statisticians have criticized me for reporting p-values as p<0.05 because it is not precise.
My question is: how to deal with a p-values that are not below but close to my alpha?

Frame challenge: Focus on the data you gathered and on weighing hypotheses against each other in metrics meaningful to your domain, not on the cult of the $p$-value.
The cult of the $p$-value is a system for journal editors to avoid putting their journal's good name on too many random flukes.  Set $\alpha = 0.05$, and assume everyone works in good faith with no selection bias in the garden of forking $p$aths, and only 5% of your journal will be raising false alarms about results that are actually random flukes.1
That's all that "statistical significance" means: it's a false alarm rate under the premise of a status quo conventional wisdom null hypothesis, without reference to alternative hypotheses.  Even if the d20 is perfectly fair, you get a journal article every time you roll a critical hit of 20.
Framing $\alpha = 0.05$ terms of "we require $p < 0.05$" is just a standard API for statistical tests: the same tests, computed by the same software (or computed by the same computers, back when "computer" was a job description), can be trivially scaled to different values of $\alpha$—and, perhaps, shopped around to different journals with different levels of tolerance for flukes—without any extra analysis, because under the null hypothesis, the distribution of $p$ is uniform random in $[0,1]$.
By definition, nothing about a $p$-value in general says anything about the plausibility or strength of evidence for alternative hypotheses, at least until you do a power analysis of the specific test in question to raise true alarms about specific alternative hypotheses of interest.  For any statistical test of a sampling process under a null hypothesis, the $p$-value is uniformly distributed in $[0,1]$, but under alternative hypotheses, the $p$-value may have any distribution—including, for some alternatives, the same uniform distribution!
You gathered some data from a system; you have some hypotheses about the system, including, perhaps, some status quo assumptions about it in a null hypothesis.  If the hypotheses make quantitative predictions, you can now compare those predictions quantitatively to weigh the hypotheses against each other—for example, if you have a likelihood function for two hypotheses, you can compute an odds ratio for the hypotheses to measure the strength of evidence for one over the other, and use that to guide a process of model selection.  You might have to meet a $p$-value threshold for publication in a journal you want, but the substance of your research is about the data, methodology, and competing hypotheses.
What distinguishes metrics like odds ratios from $p$-values is that odds ratios are affected by both hypotheses you're examining, whereas $p$-values by definition are meaningful only to a null hypothesis.  Now, it may happen that the computation of a $p$-value for some statistical test, $p = F(X)$, is also a useful function in the analysis of alternative hypotheses measuring the strength of evidence over the null hypothesis: you might happen to reuse the same function $F(X)$ of the data $X$ in the computation.2  But that's an accident of the test in question and must be phrased in specific terms of that test to have any meaning.  A $p$-value on its face confers no strength of evidence; it's only a standard interface to let editors control the fraction of random flukes in their journals.

1 Of course, you might use the same theory for purposes other than a journal editor vetting paper submissions, but the decision theory of this rule in isolation works the same in whatever domain you apply it to.
2 This is what motivated Fisher's approach of publishing specific $p$-values and treating them with any kind of meaning about strength of evidence.  However, although Fisher's method of combining $p$-values of several independent tests into a composite test with the same "statistical significance" (false alarm rate) is technically correct (as is the method of picking one $p$-value and discarding the rest), $p$-values as strength of evidence for alternative hypotheses are incommensurate and there is no general logic to reconcile them.  Fisher attempted to create a theory of fiducial inference to address problems like this, but it is largely incoherent and has been abandoned by the wayside in the history of statistics.  Unfortunately, vestiges of Fisher's abandoned theory remain in popular wisdom and textbook intellectual copypasta.
