Let $\mathbb{F}_q^k$ be a $k$-dimensional vectorspace over a finite field, and let $l \ge 0$ be an integer. The question is how to construct a (maximally) large set $A \subset \mathbb{F}_q^k$ such that if you pick any $k+l$ distinct elements from $A$, then these points span $\mathbb{F}_q^k$.
This question is a more general version of my previous question, which handled the case $l=0$.
In more geometric terms the question sounds like this: Construct a large set of points in $\mathbb{P}^{k-1}(\mathbb{F}_q)$ such that no $k+l$ lie on the same hyperplane.
In the answer to my previous question it was pointed out that for the case $l=0$ these objects are called $\mathcal{K}$-arcs and they are well known.
I believe for $l>0$ these objects are called $(\mathcal{K},k+l)$-arcs, but I could not find any construction for them.
As you've mentioned, this is a natural generalization of $\mathcal{K}$-arcs. Apart from few particular cases, not much is known about these objects. Most of the results/constructions in the literature are still dealing with the $\mathbb{P}^2(\mathbb{F}_q)$-case. It seems that, even in this case, general sharp bounds are somewhat lacking.
A simple procedure to construct a "large" set of $n$ points in $\mathbb{P}^2(\mathbb{F}_q)$, with no $r+1$ in the same line is:
Pick an irreducible projective curve $\mathcal{C}: f(x,y,z)=0$ of degree $r>1$ with a "large" number $n$ of $\mathbb{F}_q$-points. Then by Bezout's theorem, this set has at most $r$ collinear points.
To illustrate the case $(k,l)=(3,1)$ in your notation, just take an elliptic curve $\mathcal{C}$ attaining the Hasse-Weil-Serre bound: $n=1+q+[2q]$.
For additional details (bounds/constructions), you can look at the paper (and references therein): http://www.sciencedirect.com/science/article/pii/S1071579705000250