PRF that can be distinguished after $k$ queries? In adaptive attacks, if we design poorly, adversary can modify his queries and break the given pseudo random function (that is being able to distinguish it from uniform randomness).

Is there a poor way to design a PRF such that in one round of query, the adversary cannot distinguish but in two rounds the adversary can?

How about adversary failing to break at $k-1$ queries but breaking at $k$th query?
A $k$-wise independent hash family has the property that the joint distribution of $h(x_1), h(x_2), \ldots, h(x_k)$ is uniform when $h$ is chosen uniformly from the family (and $x_i$'s are distinct). Such families exist unconditionally (for fixed $k$), are efficient, and satisfy the condition you give.

The simplest example of a 1-universal function is $F(k,x) = x \oplus k$. Clearly the output of $F$ on one point is uniform (it's a one-time pad) but with two queries $F$ can be easily distinguished from a random function.